Of Particular Significance

Mass-ive Source of Confusion

POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 07/11/2013

One of the challenges for a person trying to explain physics to the non-expert — and for non-experts themselves — is that scientific language and concepts are often frustratingly confusing. Often two words are used for the same thing, sometimes words are used that are fundamentally misleading, and often a single word is used for two very different but related concepts. You’d think we’d clear this stuff up, but no one has organized a committee dedicated to streamlining and refining our terminology.

A deeply unfortunate case, the subject of today’s post, is the word “mass”. Mass was confusing before Einstein, and then Einstein came along and (accidentally) left the word mass with two different definitions… both of which you’ll see in first-year university textbooks. (Indeed, this confusion even extended to physicists more broadly, causing the famous particle physicist Lev Okun to make this issue into a cause celebre…) And it all has to do with how you interpret E = mc² — the only equation everybody knows — which relates the energy stored in an object to the mass of the object times the square of the universal speed limit c, also known as “the speed of light”.

Here are the two possible interpretations of this equation. Modern particle physicists (including me) only use the first interpretation. The purpose of this post is to alert you to this fact, and to point you to an article where I explain more carefully why we do it this way.

Interpretation 1. E = mc² is true only for an object that isn’t moving. For an object that is moving, E is greater than mc². Energy and mass are not at all the same thing; an object’s energy can change when its motion changes, but its mass never changes. This notion of mass is sometimes called “rest mass” (since it’s related to the energy stored in the object when it is “at rest”) or “invariant mass” (since it doesn’t change when it is moving.)

Interpretation 2. E = mc² is always true, for both stationary and moving objects. This can be viewed as saying energy and mass are essentially the same thing. [Recall that in interpretation 1, they are not at all the same thing.] Since the energy of a moving object is larger than when it is stationary, that means, similarly, that its mass is larger when it is moving than when it is stationary. This notion of mass is sometimes called “relativistic mass”, in honor of Einstein’s revolutionary notions of relativity.

To sum up — relativistic mass depends on how fast an object is moving, but invariant mass/rest mass is the same whether an object is moving or not; you can see this in the figure below.  Which one of these should we call “mass”, with no modifier?  Unfortunately, that’s up to the user.

The two interpretations of  what "mass" means.  The first (in green) is that mass is something that does not change with speed -- often called "invariant mass" or "rest mass", it is used by particle physicists.  The other, "relativistic mass", is just energy divided by c-squared. Note the two are almost identical at small velocities and so are essentially equal in daily life.
The two interpretations of what “mass” means. The first (in green) is that mass is something that does not change with speed — often called “invariant mass” or “rest mass”, it is used by particle physicists. The other, “relativistic mass”, is just energy divided by c-squared, and grows with speed. Note the two are almost identical at small velocities, and so are usually equal in daily life.

Fortunately, in daily life, these two concepts are almost identical, because most objects we observe in daily life much more slowly than c, in which case their rest mass and relativistic mass are nearly identical, as you can see in the figure. But particle physicists and nuclear physicists and astronomers, among others, often have to be more precise. And when you’re reading an article or book about particles or nuclei or astronomy in which “mass” plays an important role, you will often need to know which of these two interpretations is being used by the author!

Einstein, in his early years, contributed to the second interpretation, perhaps inadvertently.  But later he made clear statements (most notably in a letter to Lincoln Barnett, which I can’t find in full on the web, but which is quoted widely) in favor of the first  interpretation. Not that Einstein’s opinion particularly matters; in science we respect our elders, but we do not slavishly follow them, the way people used to follow Aristotle. We come to a conclusion based on what we know, and often we know things that weren’t known to the previous generations.  So why do particle physicists today choose interpretation 1? I’ll give you a couple of quick hints as to why, and if you want to learn more, you can read my article on the matter.

  • If you use the first definition,
    • all photons (particles of light) have zero mass,
    • all hydrogen atoms have the same mass: 0.938 GeV/c²
    • all electrons have a mass which much smaller than that of a hydrogen atom.
  • If you use the second definition,
    • no photons have zero mass (because all photons have energy)
    • every hydrogen atom has a different mass, depending on how fast it is moving; and
    • any particular electron may have a smaller or larger mass than any given hydrogen atom, depending on how fast each of them is moving.  [For instance, an electron emitted by a decaying Higgs particle has a larger mass than the hydrogen atoms in your body (at least from your point of view).]

So with the second interpretation, you can’t even say which types of particles have larger masses than other types, and it is impossible for any type of particle to be massless.  This is very inconvenient — one might even say, ridiculous — for doing particle physics.

Additional subtle but profound mathematical reasons [having to do with the “hyperbolic geometry of space-time”, if you must know] support the first interpretation, as hinted at in this article on mass and energy, which shows how energy, momentum and mass are related by elegant equations if you use the first interpretation.

Anyway, as long as you are aware of the existence of these two different interpretations, you will usually be able to tell which one is in use by an author.  On this website, the first interpretation is always used.    If you would like to learn more about why particle physicists choose the first interpretation, click here for my more detailed article.

Share via:


224 Responses

  1. Sweet blog! I found it while browsing on Yahoo News. Do you have any tips on how to
    get listed in Yahoo News? I’ve been trying for a while but I never seem to get there!
    Appreciate it

  2. Hi there very cool website!! Man .. Excellent ..
    Amazing .. I’ll bookmark your site and take the feeds also?
    I am satisfied to find numerous helpful info here within the
    publish, we’d like develop extra strategies on this
    regard, thanks for sharing. . . . . .

    1. Very interesting and accurate statement
      Time dilation is a permanent feature of physical reality because velocity v is never exactly zero because measurement error is always non-zero–namely v is never equal to 0+_0—
      Because of the very same reason rest mass is never a physical reality –namely mass can never be measured at exactly zero velocity

      Very nice

  3. Dear Prof. Strassler, Dear All

    With your kind permission,please let me state my point of view on the relationship between mass and energy—-hoping that it will not augment the apparent confusion.

    –This issue is something that belongs to physics,not just to particle physics
    –Nothing changed since Einstein ,more than 100 years ago,stated that mass of a body is a measure of its energy content
    –More precisely,c squared is the proportionality constant between the two
    –Mass is inertial or gravitational,according to the equivalence principle
    –For my present purpose, a body may be an elementary particle or a macroscopic body.
    –I assume that there is an observer,fixed in his own frame of reference
    –I assume that all physical quantities that I am talking about :mass,energy,velocity,etc are measured or calculated with respect to this frame of reference.
    –When the mass of a body is measured with respect to this frame of reference(we assume that this can be done)we get a certain value M
    –According to what Einstein said ,M is equal to its total energy content divided by c squared,
    –Its total energy content is equal to the algebraic sum of all its energy components:rest mass energy(M0 which is invariant, times c squared) ,motion or kinetic energy which is M0 times(gamma-1)times c squared ,internal energy or self energy and finally potential energy which depends on the potential at body location (gravitational,electromagnetic,etc) and on body characteristics :mass M'(M’ is M minus the potential energy contribution),charge,etc,Quite often the potential energy is negative which means that it diminishes the finally observed value (M being smaller than M’).

    NB–When velocity is extraordinarily close to c or equivalently when physical quantities are near extremum values(probably -?-close to Planck values) then body behaviour may be different,but this is still in speculation status.

    1. My humble view…
      /internal energy or self energy and finally potential energy which depends on the potential at body location /.

      Internal energy is time dependent, may be non axiomatic quantum action “h”, which may be outside the notion of mass.
      There is nothing called pottential energy at body location, because mass is relative.

      It is all the problem of quantum mechanical definition of uncertainty principle. There is no continuous virtual particles or probability density. Virtual particles can only help micro electronics.
      The reality is “relative mass” which is the position and momentum of particles and macro objects – which cannot be calculated within our conscious world.
      Through “Equivalence principle”, relative mass is adjusted as rest mass to fit our conscious world. At high energy (we may call at speed near c), the division of space inbetween quanta is reduced (reduce symmetry breaking), thus makes the matter disappear and slowdown the time (slowdown space expansion).

  4. I read your comments on relativistic mass. You made a lot of errors in it. You gave the impression that Einstein didn’t approve of relativistic mass where in fact he did and not only did he approve it but he used it in his GR text “The Meaning of Relativity” on page 102 when he speaks of the inert mass of a particle in a gravitational field. He also used it in his 1906 “The Principle of Conservation of the Center of Gravity and the Inertia of Energy,” Albert Einstein, Annalen der Physik, 20 (1906): 626-633. In it he assigned mass density to an electromagnetic field. In his GR paper he explains that mass finds its full expression in the energy-momentum tensor, which is quite true. You’ve also confused relativistic mass with energy. That’s not true either. The rel-mass of a particle in a static E-field has potential energy. The total energy is conserved but the mass-energy is not. If the particle is in free-fall in a gravitational field then the energy is the time component of the momentum 1-form, not the 4-momentum. When stress is introduced into a body its no longer isolated and cannot be described using a 4-vector. The rel-mass is then no longer related to its inertial mass through E = mc^2. There are a lot of other problems in your website but these are what pop out at me. A

    By the way. A friend of mine is a particle physicist and he uses relativistic mass. His name is Alan Guth. I’m sure you’ve heard of him.

    1. I know Alan, of course. I have not seen him use relativistic mass in any of his papers; the inflaton has a mass, and its mass is its rest mass, not its relativistic mass. I also know Nima Arkani-Hamed; he does not use relativistic mass. I’m sure you’ve heard of him too. What’s your point? You will never see relativistic mass in any LHC-related scientific paper; if you disagree, find one.

      The reason to use “rest mass” is that it is an invariant. It doesn’t depend on all this other stuff. All the other discussion that you’ve introduced about relativistic mass is exactly why it is a bad idea to use it when explaining things to the public. Of course it doesn’t matter what terms you use as long as you get the math and physics right in the end; but there is a pedagogical issue here.

      It is possible I have errors about the history of Einstein’s various ins and outs with the notion of mass, because he did take different points of view as he was developing the notion. But you didn’t read what I wrote very carefully. I did not say Einstein did not approve of relativistic mass at various times in his life; I said he used both at different times. Late in his life he made a statement that using it was a bad idea. But it doesn’t matter what he said or thought; it matters what we should do, now, today.

      As for saying I have confused relativistic mass with energy: please define relativistic mass precisely, preferably by email so we can do a proper back and forth. When the definition is precise, we can have a meeting of minds; but at this point, I do not understand precisely what definition you’re using, so I can’t answer you.

      1. ‘there is a pedagogical issue here’
        This is kind of my problem. I have no issue with people using a simpler interpretation to more easily explain or accomplish things,

        but these two interpretations:
        ‘energy and mass are essentially the same thing. [Recall that in interpretation 1, they are not at all the same thing.]’
        are clearly mutually exclusive.

        I think its important to point out what is a closer representation of reality before you then apply a simplification or we ignoramuses will start to build analogies on analogies. Which may be why I can not reconcile the idea that the mass of a Proton is both sub particles and energy, with an interpretation that deems mass and energy to be ‘not at all the same thing’.

        I looked at the mass-energy page which seems to tell me that mass and energy are indeed the same using the mass of the system calculations. Which again indicates that interpretation 2 is actually true to life with 1 being a simplified interpretation for calculations. This is also indicated in the Higgs and Gravity article.

        All of which says to me that Okuns paper use of ‘taken with a large grain of salt.’ is wrong. Its a needlessly harder way to work but actually more technically accurate.

        To put it another way, its really confusing to me to say that in one context mass and energy are identical (Mass and energy of Higgs decay taken as a system) and in another it isnt (Mass and energy of the only two components in that system) Rather than say mass and energy are actually the same but we use a simpler interpretation for easier calculations.

        So confusing I still dont really know the truth of it. Cant shake the feeling Im talking utter gobbledygook.

        1. Sorry that you are still confused. It *is* potentially confusing. It’s really critical to be precise. And it is possible that things I have written at different times use wording that is slightly contradictory, which would be unfortunate. I’ll try to find time to double-check.

          I never said that mass and energy are not related to each other. I said they are not the same.

          As of now, the clearest statement I have been able to make about the definition of mass used by particle physicists is:

          1) take the system of interest (whether electron, proton, rock, or hamburger)

          2) move it away from everything else

          3) make it stationary (to be precise: make its total momentum zero.) If the mass of the system isn’t zero, then you can always do this (or you yourself can move so that, from your point of view, the momentum of the system is zero). If you can’t do this, that means the mass of the system is zero.

          4) measure the energy of the system, including all types, including motion energy and mass energy of its pieces (if any), any interaction energy among its pieces, etc.

          5) divide the energy by c^2: that is its masss.

          Note that (4) and (5) tell you the mass and energy are closely related, but only if (3) is also true. If (3) is not true, then mass and energy are different.

          So the condition (3) has to be always kept in mind. For the definition of mass used by particle physicists, the mass of the system and its total energy are not related by E = mc^2 UNLESS the system as a whole is stationary (i.e. has total momentum equal to zero.)

          I think all of your confusion comes from not remembering to check whether (3) is true.

          1. Hmm, so the reason why the Higgs decay goes from 126GeV/c^2 to 0GeV/c^2 is because the Higgs was a stationary particle then the photons are in motion and so dont fulfil 3 and cant reasonably have a mass energy calculation done on them?

            but if the mass of the system remains constant and directly connected to the energy doesnt that imply that the photons have mass as they are the only components of that system? Isnt the fact energy is mass just being put aside for the purposes of calculation?

            Its that that makes Okuns paper and the statement that mass is not the same as energy confusing to me. I understand this methodology makes dealing with particles hugely less complex but it surely cant be how reality is working, can it?

            1. Let’s consider Higgs decay to a bottom quark and a bottom anti-quark; those each have mass of about 5 GeV/c^2. View the Higgs as initially at rest from your point of view.

              So the sum of the masses of the particles in the system drops from 126 to 10. Even though the bottom quark and antiquark are in motion, we can calculate their masses, in two possible ways.

              The first way is to send a person along with the bottom quark, so that he or she views the bottom quark as at rest. That person will report the energy of the bottom quark is 5 GeV, and so we will know the bottom quark’s mass is 5 GeV/c^2.

              The second way, is to measure the bottom quark’s energy, which is 126/2 = 63 GeV, and to measure the bottom quark’s momentum, which is slightly less than 63 GeV/c, about 62.8 GeV/c. Then we use the formula

              m = square root of (E^2 – [pc]^2)/c^2

              That will also give us 5 GeV/c^2 for the bottom quark’s mass.

              We’ll get the same answer for the bottom antiquark.

              However, the mass of the quark-antiquark system is different. Its total energy is 126 GeV. It’s total momentum is ZERO, because the quark’s momentum is in one direction and the anti-quark’s momentum is in the opposite direction. So the system is at rest, and its mass is 126 GeV/c^2, as it was initially.

              The mass of the system is not the sum of the masses of the objects that make up the system. If the system has zero momentum form your point of view, the mass of the system is the sum of the energies of the objects that make up the system, divided by c^2.

              The math works. If your intuition for the words disagrees with the math, change your intuition.

          2. Oh Id never dispute the maths. I just see that the two interpretations are mutually exclusive in what they say about what is happening in reality (Mass is energy, mass isnt energy at all.) and this and Okuns paper consider interpretation 2 to be incorrect. Okun wants it dropped entirely but I cant see any way of Interpretation 1 being as true to life as Interpretation 2 regardless of the easier maths of 1.

            From what I gather so far, much of the mass of the system becomes energy in the system and that energy is now fundamentally different to mass in that context? So a proton has two distinct masses a system mass and a component mass.
            I still dont really understand how that switch would occur in a real life system. Where does the mass go from one context to the other if it is no longer equivalent to energy while looking at the components?

            Something that might help me. In the speed example, if a ship attempts to accelerate up to light speed then presumably in Interpretation 2 they would become so massive it would collapse in to a black hole? If energy is not mass in Interpretation 1 then what happens? (Viewed as a system the results would seem to be the same as Interpretation 2 but viewed as a component nothing within the system is anywhere near massive enough.)

            I hope I am not being too frustrating and perhaps I am simply reaching the limits of my ability to comprehend this.

              1. Im sorry if I am being dim but Im afraid Im not sure how the comment solves my issue.

                I think I understand the definitions, its how they are put together and what contexts they are used in that raises some questions for me.

                1. I apologize for bothering you
                  Please kindly feel free to ignore my comments
                  However,if you still think I may be able to clarify things for you,please kindly feel free to address me with any specific question and I will try to do my best. Abraham

          3. I believe I have worked out how to get this in to my brain.
            I gave a previous example of accelerating to high speed turning you in to a black hole, which was an entirely incorrect assumption. However, I think it put me on to the right track.

            Apparently, while a fast moving object may appear to be a black hole from some reference frames it has to be like that from all reference frames to truly be considered a black hole. So thinking about it from the point of view of a hypothetical 2 particle true black hole just coming in to being, I figured that if you ever viewed the situation from one particle the other particle would contain all of the mass needed to still drag everything inevitably inward and vice versa.

            So, applying that to the example here, if you have the 126 Higgs break down in to two quarks, then bring one quark to rest and measure its rest mass of 5, if you then measured the relativistic mass of the other quark then presumably its mass would be 121.

            I understand this is probably an incredibly round about way of going about this, and I believe your second way of measuring the rest mass implied this all along, but is that right? Have I finally managed to grasp where the mass is going?

  5. Interpretation 1 : Mass means, which affects spacetime. The “invariant mass” has no momentum – which cannot affect spacetime – thus have no meaning as mass. Every mass is relative, it appears constant (rest mass) due to Moment of inertia – with each other as reference frame.

    The energy spectrum of a bound state is discrete (appears rest mass), unlike the continuous spectrum (rest mass is blurred) of isolated particles. In Quantum field theory(QED), the notion of a force-mediating particles comes from peturbation theory, and does not make sense in the context of non-peturbative approaches to QFT.

    From the point of view of quantum field theory, particles are identical if and only if they are excitations of the same underlying quantum field. Thus, the question “why are all electrons identical?” arises from mistakenly regarding individual electrons as fundamental objects, when in fact it is only the electron field that is fundamental.
    So there is no rest mass without reference frame ie.. relative mass.

  6. Dear Prof,

    I fully agree with you that your Interpretation 1, “E = mc² is true only for an object that isn’t moving” is correct.

    However, why can’t we see it in a more simple way i.e., since

    E = mc ^2 {(1-v^2/c^2)}^(-1/2)

    With Taylor expansion,

    E = mc² + ½mv² + 3/8mv^4/c² + 5/16mv^6/c^4 ……

    In other words, even in motion, the mass do not change but the only the kinetic and some other form of energy increases with the increase in the velocity v.


    Dr Looi

  7. Only after time dependent mass, there is measurable “space” (space time). Only after measurable “space”, force exists – also gravitation. It’s strength may be the amplitude of its field (stuff). So energy and momentum affect the space time.

    “Force” is a Terminology of “space” – not the mass or “stuff”. So quantum action “h” is a terminology of space.

    So without reference frame, there could be no axiomatic quantum action. But space is not axiomatic and not semantic.

  8. “Today may not an evolutionary extention of yesterday.”
    In Lev B Okun paper, he says, there is only one mass “m”, not depending on reference frame. Relativistic mass and rest mass are misleading.
    Einstein said, there can be no clear definition can be give for moving body mass – instead of giving its name as “rest mass”, it is better to mention, “momentum and energy”.

    My notion is.., “there is no terminology called mass, without reference frame” !

    Matter is a stuff, particles and fields are stuffs – but “mass” is not a stuff.
    Stuffs are “Time dependent”. Mass and energy are not the same – mass carries energy.
    If “mass” become time dependent, energy is conserved – because of inertia, momentum also conserved. There is, time dependent and time independent mass.
    Einstein said “relativistic mass”, where mass become time dependent. In this stage measurable “space” become crucial. It is the “rest mass” but could not be contained within the definition of “m”.
    The difference between “quantum mechanics” and “relativistic theory” lies here.
    Without “rest mass” quantum mechanics (“ball” on the spring) is confusing. Without rest mass, relativistic theory also confusing, but can survive with energy and momentum. This is the uniqueness of relativistic reality.

    Weathers are unpredictable and changing, but seasons are predictable and constant. Did we believe, weather will never alter the seasons ?

  9. Okun seems to want mass and energy equivalence to be dropped entirely and it really is far more confusing to me than any of the variations on E=mc^2

    Unless I am just getting this increasingly wrong, isnt the Protons mass determined by its constituent particles _and_ energy?

    Wouldnt that imply that even in Interpretation 1 you need only scratch the surface to see that mass and energy are the same thing?

    I understand that Interpretation 1 is much easier and just as effective in many contexts but the same can be said of the Bohr model of the atom, it still seems reasonable to consider a more complex view that is closer to the reality.

    That is unless my knowledge about where energy is considered mass is entirely incorrect. In which case Im back to square one in knowing where the mass of a proton is.

  10. There is no absolute rest in physics (v=0+_0)because of inevitable measurement error no matter how small.
    Therefore rest mass is not legitimate part of physics
    Therefore interpretation1 is totally irrelevant in legitimate physics.
    Therefore any legitimate (=correct) TOE can only make use of definition 2

    1. I’m not sure I follow you. When using definition 1 what matters is the reference frame, which is absolute by definition. There is also the fact that definitions 1 and 2 are both compatible with the underlying mathematics in the same way that the many worlds and Schrodinger interpretations of QM are. Your comment reminds me of the assertion that c is useless because we can never be sure of the exact speed of any object.

      1. There is a minimum velocity in physics,basically depending on the minimum time interval in physics and on the minimum acceleration in physics which is inversely proportional to the mass of the universe.c happens to be the only extremum value in physics (maximum velocity).There is a minimum and a maximum to any physical quantity,but their explicit values are not known,besides c ,which has been defined to be absolutely accurate and constant from now on.We do not actually know what happens at absolute rest or at v=c

  11. I still claim that (a bit modified) definition 2 (within certain model) is FAR, FAR more better than definition 1. With definition 2, simple and testable TOE is reality.

  12. Kudzu
    Future is always different from the past
    Future (and better) theories are going to replace past theories
    NOTHING in the future is going to be the same as in the past

    1. Future theories tend not to say that past observations were different. Einstein’s explanation of gravity does not say that the gravity we have now (or in the future) is\will be different from what it was in Newton’s time. I may have a theory that all swans are white. When I see one that is black a new theory emerges. But that does not mean that the past had no black swans in it and the future is different by having those swans in it.

      Imagine a universe where the future truly was different from the past, where at some point the old order ended and a new one began. One where the very mechanisms of reality changed. Would anything survive it? Imagine if gravity suddenly became equal to electromagnetism or if protons could suddenly decay. It would destroy all that is.

      1. Thanks for your comments
        1.Of course past observations do not change.What does happen from time to time is that new obsevations are made that do not fit old theories
        2.In this case,a new theory has to be devised,that will explain both old observations and the new observation.
        This is how theories evolve and this will happen as long as there are physicists

        1. Assuming that we do not 1.) Reach a ‘theory of everything’ or 2.) Reach the limits of what we can observe and discern. (There is after all no reason that the universe need be comprehendable by a bunch or organic dross on a tiny lump of rock around a middling star.)

          1. We will never reach a theory of EVERYTHING
            We will never reach the limits of knowledge
            We will never comprehend who created the universe and how
            We will never comprehend who created life and how

            1. But how do we KNOW that? Opinions are fine but they have a tendency to be wrong.

              (Of course we can always ask ‘why?’ to any given explanation ad infititum, but what if we reach the stage where no further gains in knowledge are possible, a point perhaps where our very mental structure cannot proceed further?)

              1. Intellectual curiosity is what characterizes humans.They always ask questions .When they stop asking questions,they stop being humans.You ask how do we KNOW that.We DO NOT KNOW that.WE ASSUME that.We always assume that there is a better theory of everything, namely more knowledge out there for us to find out.If we do not assume that , we stop being humans.When we stop being humans,we start to be animals.History is full of examples like that.Our destiny is to be humans.So let us keep asking questions.Any questions.With no fear.

              2. The toughest question is WHY
                Why was universe created
                Why was life created
                Why was man/woman created

      2. Future is different tomorrow from today in the sense that the weather tomorrow is different from the weather today

        This is all I meant

        1. True, but the weather today (and tomorrow) is just an extension of the weather yesterday. The world is always changing, but it is just a reordering of the past a logical (if exceedingly complex) follow-on. We are always referring to things being the same even when they change. The cells in my body are born, die and are replaced. The atoms in me cycle through the world, yet I am still ‘me’ still the ‘same’ though so much changes the essence the ding an sich of me remains the same.

          1. I would like to think that people always want to change for the better,intellectually,behaviourly or otherwise(e.g. loose weght…)
            The only thing that doesn’t change is their I.D. or S.S.number…)

          2. Mr.Kudzu, Mr.Abraham Sternlieb’s approach soothing me. Like quanta, weather, there is void between events. “Today may be, not a historical extention of yesterday.”
            In causality, inertia, momentum, mass, speed of light, Equivalence principle.. are all – may be the effects rather than the causes. Sudden change in observable cause, can change these observable (measurable) effects ?
            Forgive me if there is any mistake.

            1. Void is not well defined(=time or space interval?)
              Event is not well defined(is it zero-sized and zero-duration ?)

            2. Interesting
              The notion of VOID (or VACUUM) is not well defined
              Is it equivalent to “time interval” or “space interval”???
              The notion of “event” is not well defined–
              Is it a zero-sized and zero-duration entity ???

        2. Void = Non-Axiomatic.

          Event = Axiomatic – in this case, almost nonrecurring weather pattern. The Massive Pandemonium in Matter (courtesy: Professor Strassler).

        3. Void = “space interval”. But not Semantic.

          We speak about change of event – which intuitively affect us.
          “Anarchism” may be a suitable word. It is an order, not a disorder.
          The causes are in disorder (Pandemonium), which tends to change. But the effects are constant (event).
          Is it a zero-sized and zero-duration entity ???
          Within my small knowledge, events are localised by constants (effects).
          Space is expanding, but our piece of flat universe is protected by constants like Equivalence principle ?

          1. In physics, and in particular relativity, an “event” indicates a physical situation or occurrence, located at a specific point in space and time. For example, a glass breaking on the floor is an event; it occurs at a unique place and a unique time, in a given frame of reference. – In which, we give birth, play, grow and die.

            1. Unique place and unique time…………………….
              No matter how small or how short in time,a physical event is never zero-sized or zero-duration because of inevitable measurement error
              Therefore a fundamental minimum uncertainty exists not only in Quantum Mechanics,but also in Classical physics
              This minimum signals approach to the Quantum Gravity limit

  13. Matt–More recently, E = mc^3……..(everything is going up these days…..)

    Doug Gluntz

  14. I also some time try to read in my textbook if there is any connection between Newtons big G and Coloumbs electric constant e0 ? And if I understand the physic community correct there is no one yet that can prove any conection between those constants, and when that connection is discovered then it would be the grand unified theory or TOE, or ?

    1. At present nobody knows why c is the same for everyone or why it has the precise value it does. We may never know, such is science. You can always keep asking ‘Why’ until you reach that point where we just don’t know.

      A TOE may arise in many ways. One of these is to unite the fundamental forces. We have managed to tie electromagnetism, the weak and strong forces together, but not gravity or the Higgs force (That I am aware.) If a connection was found this may give us a TOE (Or GUT, Grand Unified Theory) or it may just present us with more questions.

      1. Nobody knows why c has the value it has.
        This is true for many other parameters in physics,like electron mas etc
        There always will be a set of physical parameters that will have to actually be measured and all others will be derivable from them.
        Naturally,physicists’ goal is to devise theories that will minimize the number of parameters that should be actually measured
        In mathematics,you do not actually have to measure anything unless counting or calculating can be called measurement…..

        1. Incidentally,there is nothing that prevents c from changing in time,and this is true for all physical constants !

          1. Speed of light c is measurable because, we have less knowledge about “space” ? Planck mass was a wonderful thought about “invariant mass”.
            The added factor of 1/√8π simplifies a number of equations in general relativity. because the unit measures the approximate scale at which quantum effects, here in the case of gravity, become important.

            Rest mass lies in the quantisation of energy E = hv and its strength in number of quanta. Planck was skeptical about this Boltzmann assumption – but convinced by Einstein’s photoelectric effect. According to de Broglie, the neutrino and the photon have rest masses that are non-zero, though very low. That a photon is not quite massless is imposed by the coherence of his theory. In addition, he believed that the true mass of particles is not constant, but variable. De Broglie’s final great idea was the hidden thermodynamics of isolated particles.
            Einstein was skeptical about the wave nature of the matter. de Broglie said “that the particle must be the seat of an internal periodic movement and that it must move in a wave in order to remain in phase (simple harmonic motion or perpetual momentum) with it was ignored by the actual physicists wrong to consider a wave propagation without localization of the particle, which was quite contrary to my original ideas.”

            So the Geometry of space decides the mass – than the relativistic with c or the intrinsic quantities of energy and momentum.

          2. Very true, but I believe there have been measurements to see whether this is the case, notably that of the fine structure constant? This would put definite limits on how physical parameters have changed in the past would it not?

            1. The fine structure constant is a dimensionless number and has been found to be quite constant in time.However,past experience cannot say anything about the future.The speed of light,c,being of such fundamental importance,has been arbitrarily fixed at his present value(with no error attached).The standard meter has been redefined as being the distance travelled by light in vacuum in one second,divided by above-mentioned recently fixed speed of light.So,c has been declared absolutely exact and constant.If,e.g,,tomorrow c actually doubles,then the meter will become twice as long

              1. Tomorrow if c increases 5% we’ll all be dead before we find out.

                I would argue that the past is the only way we can know about the future. As soon as we measure something it is in the past. In order to get an accurate result we need record, all of past observations. Our basic functioning is based on the fact that we assume the future will be like the past, that trends will continue that the universe will be predictable based on past behavior.

                This of course may not, indeed cannot be 100% true, but seeking 100% truth is a fool’s errand and a snipe hunt.

                1. Of course anything we do becomes immediately the past.
                  However in science it is quite usual that ,from time to time,theories and facts that were thought to be “solid truth”,are shattered by new ideas,theories and facts–examples :relativity replaced Newton,dark matter replaced the standard cosmological model and many other.

                2. Indeed but none of those theories suggest the future is different from the past, instead they are the result of improved observations, better knowledge of the past and present.

                3. By the way Kudzu, I borrowed your comment about 100% truth, referencing the source of course, for an unrelated FB post to a lay audience. I felt it was a necessary bit of philosophy to apply to many of the polarizing discussions we are having in the USA. I hope you don’t mind.

                4. That’s no problem; we need that sentiment. So many ills in the world arise when we get so focused on being absolutely right that we ignore the finer points, edges and other opinions.

                5. Never can anybody be absolutely right
                  Everybody is always relatively right

                6. As I said—

                7. Your comment on c increasing by 5%—-
                  Of course you may be right
                  However we all will be dead sometime no matter what–so what ?
                  I only tried to present scientifically correct explanations whether or not this is going to be dangerous…

          3. We know Max Planck, de Broglie, Albert Einstein… were geniuses. They themselves said about constants – were based on abstractly defined structures (Axiom).
            Why they said that is explained by Mr.Abraham Sternlieb – “Incidentally,c was measured to be the same in all frames of reference.”

            Because everything is revolving (angular momentum), our “localised” consciousness get nausea, if we take into account this reality in our daily life. So our mechanical understanding of “forces” not compatible with this reality. Relativity solved this problem.
            Genius means, they offered something more than this logic. Einstein was sceptical about matter wave. Even de Broglie conservatively said, “ball (localized particle) on the spring (wave)” – not wave nature of matter.

            But why they opted for curved spacetime, which is more of waves and fields – than to prove invariant mass is an intrinsic property of matter. Is that only a mathematical (geometry) extension of relativity or towards any measurable “physical” property of “space” ?

            1. Please kindly try to formulate more clearly your question and I will try to answer you as well as I can

          4. Speed of light is a constant. But it cannot hold the uniformity of inertial frame of reference – nor the energy and momentum. Because, from particle to planet, there is perpetual momentum – is constant with reference to each other, as if stationary on earth. Making no difference between Newton’s law and relativity.
            Who holds this alliance ? spacetime ? Dark matter pull ?

            1. Today c is defined to be of a specific value,constant in time and absolutely acurate–this is a very special occurence in physics,based on the believe that c will not change significantly in the foreseeable future
              Thanks for your interesting comment

        2. Sorry, without geometry, there is no spacetime. Space and time are independent philosophies – as you said.
          The curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present.
          Under relativity, all of these parameters are variables. Einstein field equations equate local spacetime curvature (expressed by the Einstein tensor) with the local energy and momentum within that spacetime (expressed by the stress–energy tensor).

          There is no evidence of mass here, as an intrinsic property of matter (invariant mass). Because there is concrete geometry, energy and momentum create spacetime curvature – and thus mass ?

          1. The renown relativity physicist J.A.Wheeler said : “Spacetime tells matter how to move,matter tells spacetime how to curve ”
            This is the essence of GR

            1. The eqivalence of inertial mass and gravitational mass is the fundamental basis of General Relativity

  15. Kudzu: Thank you so much ! It was a very good answer. It both was pedagogical and in accordance with the textbooks that I have. Even tough I now got your precise explanation and have read the textbooks on the subject, I just still is not satisfied. My brain should like to know the reason, the why, why C is as it is for every observer ? For instance: Is it a refractive feature of vacuum space that is physically causing that dilation / lorentz transformation ? Or is it something else in it ? I am getting crazy of this kind of subject but it is probably me that is not sharp enough. Sorry and LOL at the same time 🙂

    1. C is the speed of photon. Only thing changing is its frequency depending on gravitational interaction. Why? you may ask. Current mainstream physics won’t give you the answer.

      1. General Relativity gives you an answer,e.g.the IR gravitational redshift,namely frequency diminishing in a gravitational field

          1. A simple explanation—————-
            When a photon is emitted to outer space from the surface of a star,it has to overcome the gravitational field of the star .Unless the star is a black hole,a photon will succeed in leaving the star ,but will have a diminished energy,by an amount equal to the escape energy from the star,which is determined by star’s mass and radius.A photon with a diminished energy has a diminished frequency,namely increased wavelength,and this is called a redshift
            Actually you do not have to go through General relativity,but you can use special relativity and the equivalene principle .You take the usual SR gamma,and replace in it v by the escape velocity.Then the redshift is given by
            1+z=gamma, where z is the redshift amount,percentage-wise

  16. If a bunch of particle is moving with a velocity of 0,999999 C relative to the cern physicists in Geneva: Why does time slow for the particles ? I know that this is mathematically explained by the Lorentz transformation. But what physical process causes the particles time to slow relative to the Cern physicists ?

    1. Your question as asked does not quite make sense; time dilation IS a physical process and your question thus reduces to something similar to ‘when I push something what is the physical process that makes it move?’

      However we can see why time dilation occurs with a simple example. Imagine that you are at CERN with a bunch of particles moving at 0.866c (For simple math) and you have a laser and mirror moving with you at the same speed. You the fire a laser west at the mirror which is 1 light second away. The laser photons move there and back, a total distance of 2ls taking 2 seconds to do so. Good.

      At the exact same time you are headed towards me. I am 1.73 light seconds south of you and also timing your mirror experiment. From your viewpoint you reach me in 2 seconds (1.73/0.866 = 2) at the exact same time the mirror photons get back to you. Also good, what’s the problem?

      But when I look at your mirror experiment I see you, the particles, the the mirror, all moving towards me at 0.866c. The laser beam however is also moving and I see the beam move along a ‘<' path, first southwest then southeast. In fact I see the beam travel a total of 4 light seconds by the time you reach me.

      Now you might say 'So what? You see the laser photons move 4 light seconds in 2 seconds. The beam seems to move at 2c doesn't it? And that is what appears to happen in normal life in similar situations with physical objects.

      But we have done experiments like this and this is the cold, hard, incontrovertible fact: *I see the laser photons move at c, NOT 2c!* To me the laser photons take four seconds to bounce between the mirrors and you take 4 seconds to get to me! (This also means that to me you were 3.46 light seconds away when we started.)

      So what happened? I say that your clock is running slow (after all, *I* am always right!) You just think it took 2 seconds because of that.

      You on the other hand say that I got my timing wrong because I was all squashed up. all my distances towards you were only 50% what they should have been (Length contraction.) It had to be my fault because you're always right.

      So which one of us is correct? Did your time slow down or my distances get squashed? We can't say, that's relativity for you. If you want a 'physical process' behind those effects it's this: Light always moves at light speed for everyone all the time everywhere anywhen.

      1. A SHORT ANSWER——
        Time does not go more slowly—-Time intervals become longer(dilation)
        Space does not contract—space intervals become shorter
        Time and space are philosophical terms
        Time and space INTERVALS is physics
        All this happens because c is the maximum speed in physics

        Time and space intervals is physics
        All this happens because c is the maximum speed in physics

        All this happens because

        1. If that is physics then it is certainly not the same physics that is described in ny textbook called “Advanced physics” published in Oxford university press and authored by Steve Adams and Jonathan Allday, page 368! The conclusion there is that “moving clocks run slow”. That book cite the observation of high speed muons that has longer half life caused because their time slows in high speed!

          1. Muon half life (or in short “lifetime”) is the time interval between the event we call muon creation(birth) and the event we call muon decay(death) .
            This time interval when measured from an earthbound lab(fixed on earth) is 20 times longer than the same(similarly defined)time interval when measured from a frame of reference moving with the muon.
            This is the reason we detect muons at sea level ,because the extra lifetime allows them to do that.Otherwise they would have decayed immediately after their creation by the impinging cosmic rays,,at the top of the atmosphere.
            So,forget about time and clocks,it is only time intervals between two events that matters

            1. I am not sure that reply makes logical sense. I think what you are saying is that saying ‘clocks run slow when moving’ is not correct because the muons themselves are not clocks and that instead we can simply talk about the muons taking a longer time to decay from an earthbound perspective than from the muons rest frame.

              But we have accelerated atomic clocks and observed that they fall behind similar clocks kept at rest, so the slowing is variable; a given physical process can take different amounts of time from my viewpoint even though the system itself is not changed except relating to speed.

              In that case what is the difference between saying ‘things take more of my time units to be done at speed’ and ‘a moving system’s time units become longer at speed’? (‘Time runs slow’) Doesn’t it just become semantics?

              But that is equivalent to saying that time is slower for them

              1. Of course semantics is important.
                I just wanted to clarify the issue of contraction and dilation from the purely physics point of view.
                As nice as it sounds,but terms like : space,time,flow of time,nature,reality,etc–are Philosophy,or literature,or art,etc
                Physics contains only things that can be measured like time and space intervals which are differences between two readings
                All you have to do is to use identical clocks in all your measurements,in rest frames or in moving frames

                Incidentally,c was measured to be the same in all frames of reference.
                Einstein made this fact into a postulate (axiom).
                One way to somehow “understand” why c is invariant,is to accept(assume) that c is the maximum possible speed in physics.If you do this then by the principle of SR you obtain deductively the invariance of c.(simple logical arguments)

          2. Thank you Mr. Abraham Sternlieb
            /Einstein made this fact into a postulate (axiom)./ — The questions which haunted me is well explained !

      2. If only light speed were infinity, it would all make perfect sense. Oh, and please tell Planck his constant turned out to be zero.

        Then we can go back to working on things we understand. End of fire drill, ha ha!

        Seriously, no photons were harmed in the writing of this comment.

        1. It is dearly to be wished; we could then see the edge of the universe (taking into consideration a few effects.) Think of the knowledge we would gain.

        2. In physics no quantity (including speed and planck’s constant )can be infinity or exactly zero because physics deals only with measurable things.
          Zero sized and infinite sized things cannot be measured
          Therefore they do not belong to physics(maybe to mathematics…)

    2. Guest, that is an excellent question (and don’t let anyone tell you otherwise). Prediction: You will not receive a single satisfying answer – because there is none. The only answer that can be given by the theory is that since under the theory the critical parameter is the velocity relative to the observer, then the CERN physicist as the observer will experience a universe where time slows for the particles.

      To me the theory is like the old wives tale “a watched pot never boils”. You are the person reasonably asking: “If so, why would the act of merely watching prevent boiling.” Don’t expect a satisfying answer. Perhaps someday a more satisfying theory will come along, or an experiment will be performed that exposes the “watched pot theory” for what it really is. Until then, expect smoke and mirrors…

      1. Would it be reasonable to hypothesize that c is given inside an atom? If it’s given inside an atom then something inside an atom makes c as a top speed.

        1. Who knows? The theory is an important part of 20th century (and now 21st century) physics. I will be frank. No alternative hypothesis will supplant the theory until an experiment is performed that blows it away.

          I would suggest (and actually have suggested) performing the Michelson-Morley Experiment at orbital velocity (17,500 MPH) on the International Space Station (ISS) to see if the same null result obtained when the experiment is at rest relative to the Earth is obtained when it is moving at a decent clip through the Earth’s gravitational field. Perhaps the critical parameter is not the velocity relative to the observer, but the velocity relative to the dominant gravitational field. Such an experiment has never been performed.

          1. What interests me is the astronomical results of such a possibility; any massive fast-moving object creates a moving gravitational field; would your effect be apparent with the galaxies in the universe that are moving away from us (Or in case the expansion of space does not count, those moving through space fast enough to be moving towards us.)? Binary star systems, especially those involving neutron stars (We now have quite a few some of whom orbit each other in days or hours.) create rapidly moving gravitational fields as well; shouldn’t it be possible to examine light from such systems and look for deviations from what is expected?

          2. I actually believe that there is an experiment (Modified Cavendish Experiment) which is the key into TOE. The experiment was done few years ago but the person/team who did it couldn’t find the right explanation.

      2. I will repeat, we have done experiments many, many thousands of them. A moving clock slows, this is not just the result of a theory dreamed up but solid evidence. Any theory that comes later will have to explain these observations, it cannot simply state ‘time dilation doesn’t happen.’

        1. Yes Kudzu, a moving clock slows (who said, or even implied, it didn’t?). But moving relative to what? Few people ever bother to ask that question. Guest did, and it is a perfectly good question. Is it really the clocks motion relative to the observer – the CERN physicist – that causes the clock to slow? Or is it the clocks motion relative to the Earth’s gravitational field (in which the CERN physicist and CERN itself happen to be at rest) that is the real cause for the slowing? Frankly, the second is far more believable, at least to me. Also the second is not prone to the same paradoxes as the first, but that is another story…

          1. Clocks do not slow,they just indicate time.Only time intervals change .
            All clocks,in all frames of reference,are assumed to be identically built
            As I wrote in some other comment,clocks measure time intervals betwee two events(namely differences between two readings
            Time intervals are longer between two specific events when measured by a clock moving with respect to those two events,compared with time interval between the same two events when measured by a clock at rest with respect to those two events
            There is no simpler explanation

          2. Another way
            Nohing happens to the clocks–it is only the time dimension that stretches out…the space dimension contracts,,,

          3. True, but the first explanation holds true for satellites above the earth where gravity is markedly weaker and the effect as far as I am aware does not seem to be affected by the moon or sun, which effect significant changes in the gravity of any spot on earth. It would also affect the gravitational redshifting of light and would result in the speed of light not being the same for every observer. (For one thing there would be an Ultimate Rest Frame, that of moving at any speed away from any gravitational field.) As interesting an explanation as it is I do not think it sits with experiment.

          4. Kudzu, you are correct, the second hypotheses does imply the existence of a locally preferred frame. Are you sure there isn’t one? Suggest you get a hold of a 1971 experiment by J.C. Hafele and R.E. Keating involving around the world atomic clocks. Basically the experiment involved 3 atomic clocks – all initially set to the same time. One at rest on the ground, one on a jet traveling west, and one on a jet traveling east. Based on their knowledge of special relativity the experimenters initially assumed that when the effects due to General Relativity were subtracted out they would find the two traveling clocks ran slower than the one on the ground.

            Much to their initial surprise they found that clock moving East slowed the most, the ground based clock was next, and clock moving west slowed the least! Numbers wise, the results were in perfect accord with Special Relativity – provided (and this is the part most people are unaware of) that a locally preferred reference frame was chosen: the center of the Earth.

  17. Thanks a lot, Alain. I would add that, though important is, we can’t every time review all single logical step of the overall reasoning.
    What if every Physical Review article would adapt to a such requirement?

  18. In philosophy it existed a statement of this kind: “if two things are perfectly equal, they are the same thing”. QFT has no reason to deny electrons (or muons, or any other elementary particle you think of) to be equals, except their environmental properties like coordinates, etc. But physicist don’t believe it exists one single electron in universe. So I wonder: does QFT hides a more fundamental property of nature that accounts for this apparent violations of a philosophical principle? Will we discover some differences in mass between particles in the future? Do we have to repeat in a very sophisticated way Millikan’s experience in the future? Will electrons reveal to be elecron1, electron2, etc. ? Are environmental properties fundamental properties?

    1. Let us measure separately masses of two electrons
      Mass of first electron will be any value between M1-DeltaM1 and M1+DeltaM1 infinite number of possibilities)
      Similarly for second electron
      DeltaM is inevitable measurement error

      Therefore probability of two electrons to have identical masses is ZERO !!!

      1. Yes, of course: every theoretical physics statement loses its meaning if not rested on the solid pavement provided by specific language (and tools, and rules, and all it counts) of experimentalists. Hoc dicto, it’s highly convenient to say “electron has A mass”, avoiding “every electron has its own mass, we are unable to measure all single electron mass, so we measure some electrons’ masses, from these we argue A electron mass because we make a precise statistical inference… etc etc”.
        All that is obvious, however “of particular significance”.
        But you make me recall some thirty years ago, when as a young student at Pisa I was listening a Luigi Di Lella talk about hadronic jets from CERN’s SpS UA2 collisions (while Rubbia collaboration was in the UA1, to collide with a Nobel…). A highschool teacher asked Di Lella why trust on “detector tracing” to reconstruct submicroscopic phaenomena if traces width undoubtedly were many orders of magnitudes larger than HEP typical length. For a while embarassed Di Lella stared the teacher, although nobody would have denied a “particular significance”to that layman listener’s issue.

      2. You simply completly ignore the meaning of measurement errors and probabilities. If you measure the mass of the same electron by different methods or/and at different times, you will find any value between M1+-DeataM1. Does it mean that this very electron has not the same mass as itself ?
        Every electron in the world has the same mass, and there are very good theoretical and exprimental confirmations for that.

        1. You always have a non-zero measurement error.
          Therefore you never know the absolutely true value of anything
          You only can say that the “absolutely true value” is anywhere between two numbers
          To assume that all or any two electrons (or anything else for that matter)are absolutely identical is pure speculation or at best a totally unfounded assumption because it is impossible,in principle,to check.
          Moreover ,there are very powerful physical reasons why e.g. two electrons masses are somewhat different,because of their inevitably different “environmental”conditions at their respective locations (including all agents that give a particle its mass)

          1. Now this is interesting. While single measurements on a single particle fall prey to your objection there are several ways around it.

            Firstly we can make multiple measurements of many particles. (Or even the same particle) This will give us a distribution that gives us the expected value to increasingly high levels of confidence. This is not 100% proof, but saying ‘error just *happened* to produce this distribution that’s exactly like what you’d expect for mass m’ is guilty of moving the goalposts and the ‘God of the gaps’ style reasoning. (Measure more, I’ll be right. I’m not? Measure more!)

            Secondly, nothing can ever be proved 100%, so I may state all electrons have the same mass and you cannot prove me wrong. This puts any assertions requiring 100% foolproof assertion outside the realm of science which only deals with things that are the (often very) most likely to be true.

            Your comment on environmental conditions is interesting and it has some basis if you take ‘energy’ to mean ‘mass’ (Definition #2) however the universe itself has a ‘cutoff’ limit past which two measurements cannot be meaningfully said to be different. In that case there are only a limited number of masses any given particle may have before environmental effects become too faint to matter.

            Finally there arises the question of what ‘different masses’ means; the rest mass of all particles of a given type is believed to be identical, though their energies will likely not be. This is why two protons can be said to be identical even though one is a hydrogen nucleus and another in uranium is in a totally different environment.

        2. It is highly improbable that any two electrons have exactly the same mass,up to the last digit no matter how many digits.The more accurate the measurement methods become,the probability increases that we will be able to discern between two individual elecrons and then we will call them Electron1 and Electron2.
          Moreover,it is highly improbable that a specific electron,say Electron1,will maintain the same exact mass in the course of time,up to the last digit (see considerations above)

      3. I am not sure that’s how probability works. Measurement error is not the same as saying that the particles actually weigh that much.

    2. Firstly a philisophical principle is just that, it is not a fact or something we should live our lives by. ‘I think therefore I am’ (A rough translation) is nice but not a theory for how the brain works.

      Secondly you will find plenty of argument in philosophy about what a ‘thing’ is and what ‘identical’ is. (I am me, am I still me if one of my hairs falls out? All of them? If my head is removed? Where is the dividing line?)

      Thirdly the fact that particles differ in location is just that… a difference which would suggest that any two particles in different places are not the same. (Many bosons can be in the same place at once and be in the same state, then there is entanglement; both of these would seem to be far better examples of ‘two things actually the same thing’)

  19. Without a geometry (reference frame) there was no closed system of baryons and leptons – in the early universe – otherwise it was became undetectable (in reference with c) like dark matter.
    More momentum more mass (lot*)- so it is relativistic. #2
    Because dark matter doesn’t collide in the traditional sense, it has no way to shed its momentum and angular momentum (in the form of emitting photons) the way normal matter does.
    Focusing on the evolution of angular momentum – not only is individual particle angular momentum not conserved, but the angular momentum of radial shells also varies over the age of the Universe by up to factors of a few. It was found that torques from external structure are the most likely cause for this distribution shift.

    Frictional forces among the baryons have the general effect of removing angular momentum from baryons that have little and transferring it to baryons that have a “lot”*. Dynamical friction of dark matter on clumps of baryonic matter has the general effect of transferring angular momentum from the baryons to the dark matter.

    So the relativistic mass could be changed by the dark matter ?

  20. Thank you, Matt. That is very helpful!
    Question: Since, in the terminology you use, “mass” and “rest mass” are synonymous, do you say “relativistic mass” when referring to the larger quantity that (I think) you have to calculate so you can accurately curve the trajectory of a proton moving at very high speed?

    1. You can do. ‘mass’ is confusing in the same way ‘energy’ is. Stating either ‘rest mass’ or ‘relativistic mass’ will tell someone more specifically what you mean.

      1. If I understand him correctly, Prof. Strassler is urging us not to use the term “relativistic mass.” That is why I am asking him about a situation in which there seems to be a good reason to make the distinction verbally.

  21. General relativity generalises special relativity and Newton’s law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present.

    This means… m = E/c^2 (energy) cannot attract energy of “c”.
    Anyway momentum = c^2, mass has no meaning relative to speed of light “c” ?

    1. Gravity responds to the energy of an object even two massless particles (By definition #1) will exert a gravitational effect on each other.

      The energy and mass (or just mass by definition #2) of an object depend on its ‘rest mass’ plus its momentum. For objects that are massless (by definition #1) however this indeed becomes nonsense. However there is a second way of calculating momentum; p = hλ. That is the momentum is related to the particle’s wavelength. This works for massive and massless particles and with definitions 1 and 2 in this topic. As such we can still calculate the momentum and thus energy of particles moving at any speed.

      1. Kudzu,
        A little typo.λ= h/p .Let me say something about our blog exchange on BH. There was no reply button on that (may be that is how it is designed to prevent blogs and counter blogs!! )My understanding is that Schwarzschild solved GR equations for a spherically symmetric black hole. The solution has two singularities, one at r=0 which you have to live with. The other at 2GM/c**2 (event horizon) can be gotten rid off by changing coordinates by using Kruskal coordinates. Although there is no real singularity there, it is a place of no return. These equations are solved with respect to black hole reference frame, not with respect to an arbitrary observer. So the M is the usual mass of the system at rest.

        1. Recent (and not so recent) work has made the event horizon a much more mysterious place: http://en.wikipedia.org/wiki/Firewall_%28physics%29 I literally cannot wait to see what the results of this line of research. (Ditch the wiki page and delve into some of the sources, it’s fascinating stuff.)

          (And yes occasionally the reply button is removed when the page thinks you are spamming.)

          1. As far as I can tell,intense battle is still going on about possible firewalls, with no final conclusion in sight. In the meantime, I understand, everyone uses Schwarzschild solution or for rotating BH, Kerr’s solution for formation of BH. Hopefully Matt will write an article on black holes sometime.But to me it is clear that the mass in the equations is the rest mass. I am not sure if you can replace it by total energy or not. Perhaps you can.

            1. Well we have been informed by Mr Strassler that we can make a BH out of photons, on the other hand a collection of more than one photon has a rest mass…

              Either way the firewall tiff should have interesting results.

      2. Mr.Kudzu Thank you, by definition #1, for massless particles “invariant mass” never exists – energy is conserved in quanta and not playing any role on gravity. Only momentum is non zero.
        momentum = mass x velocity
        momentum of massless particle = zero x speed of light “c”
        momentum = zero x infinity = non zero.

        Albert Einstein had opned a window towards the reality – towards the world of everything is moving. Mass become phenomenological without a frame of reference (#2).
        p = hλ works for massive and massless particles because of photon’s particle-wave duality and – because of Einstein’s photo electric effect and quanta (E = hv). The wave nature of matter was accepted.

        Why this “relativistic mass” cannot exert gravity, because E = m ? (by definition #2)
        Only its momentum can exert pressure or resistance on constant speed of light (the realm of photon geometry). So spacetime (geometry) is strained ?
        Because, mass of photon at “rest” never exist, momentum or wavelength is non relativistic. But to calculate the electron wavelength, it was relativistic with “c”.

        1. Gravity responds to the total energy in a system, so massless particles still exert gravity. If we use the equation relating momentum to frequency then rest mass is just a minimum frequency some particles have and we don’t have to worry about E = mc^2 or its derivatives. (In that context anyway.)

  22. I noticed some people here mentioning gravity but I would like to ask my question anyway.
    What exactly does affect the gravitational force between two bodies – their invariant mass, or relativistic mass?
    I did not do the math around it so it’s all guesses but I have a feeling it’s the relativistic mass that may play role here.
    Relativistic mass has sense in this case because you can always calculate relativistic mass of an object relative to another object.
    And another observation that falls into the picture is that a black hole grows regardless whether you drop normal matter or antimatter into it, or you just shine light in there. Theoretically (or at least I heard so) it is even possible to create a black hole from just photons – and if yes then there’s a question about at what moment does that sum of photons start to gravitationally pull other objects around it? It makes a lot of sense to me to say that they always do that, even before they become a black hole.

    1. Gravity relates to the total energy content of an object, so it is the relativistic mass that matters. This does mean that light exerts a (very tiny) gravitational pull.

      1. In that case we may not need curved space to explain curved trajectory of light around a massy object. Because the reason behind the curved space argument is that light is massless and therefore cannot be affected by gravity.

        1. Good thought, half right. Most people don’t realize it, but Newton’s gravity also predicts that light will be deflected when it passes by a massive body, however the amount of the deflection is ½ that predicted by Einstein’s General Relativity (GR). In 1919 (and all experiments thereafter) measurement of the deflection of starlight during total solar eclipse has shown agreement with GR. The 1919 observations made Einstein world famous…

        2. Sure, sorry for the delay. Try: “The Physical Foundations of General Relativity” by D.W. Sciama and “Was Einstein Right? – Putting General Relativity to the Test” by Clifford M. Will.

    2. Two comments,in order to deal with these issues properly
      1.One has to define a frame of reference,relative to which all pertinent physical quantities are calculated
      2.When dealing with gravitation,one has to use General Relativity and talking about a “force between two bodies” in the Newtonian sense is misleading(in short,special relativity terminology is not relevant here)

    3. This has already been addressed somewhat, but here is another take. The short answer is that it is the relativistic mass, in addition to the momentum, that gravitates. But it is even better (and equivalent) to completely abandon the idea of relativistic mass as Matt advocates and say it’s the object’s total energy (rest mass + kinetic + potential) and momentum that really gravitate.

      First, the difference between relativistic mass and rest mass only matters for relativistic particles (moving at speeds near the speed of light). So you need to have relativistic particles for the answer to this question to really matter. Also as Matt points out you never have to think about relativistic mass at all, you can always think about “mass energy” (= rest mass * c^2) and “kinetic energy”, and really it is much easier to think in these terms.

      When you do have things moving relativistically, you find that the mass by itself is not the important quantity sourcing and responding to gravity. In intro physics we learn that mass is like the charge for gravity, but while this is a useful way to think in the nonrelativistic limit, it is incomplete relativistically. When you are dealing with gravity + relativity, you have to take into account the fact that not only the mass energy gravitates, but so does the kinetic energy AND the potential energy AND the momentum. The total energy and total momentum are wrapped up in a quantity called the “stress-energy tensor”, and this is the relativistic completion of the object’s mass in Newtonian gravity that you have to think about if you want to understand relativistic gravity.

      So photons in fact do exert gravitational forces on their surroundings, which they must since photons respond to gravitational fields. It’s not the photon’s rest mass (which is zero), but rather its energy and momentum, which responds to gravity. In fact for a photon the momentum is just as important as the energy for figuring out it’s gravitational effect. So if you want to think about it in terms of relativistic mass, you can say the photon has a relativistic mass as Matt points out above and gravity responds to the relativistic mass. However I (and basically all physicists) think it is much cleaner to say the rest mass is zero and the photon feels gravity because of its kinetic energy and momentum. It’s just a matter of accounting and what labels to give the contributions before you add them up. As you might guess, the gravitational field created by a single photon is so extremely small as to be completely negligible except as a thought experiment.

      A black hole forms if the energy density is high enough. This energy can be mass energy or kinetic energy, gravity doesn’t distinguish between different forms of energy. So indeed if you had a laser that was epically intense enough, one expects you could in principle pack enough energy into a small enough space that a black hole would form. However, to put it mildly, there are severe technical challenges to being able to actually put that much light in one place and such a thing would never occur naturally, except maybe in the very very very very very early universe (well before the point where we have any real idea about what is going on in the universe).

  23. Sorry a layman’s question:
    In the world of everything is moving, how can we calculate the distance between them (inverse square law) ?

    It has always been assumed that the parameters dealing with the Earth/Sun distance, were fixed at various precise astronomically relative positions, such as the Spring and Autumnal nodes and the Aphelion and Perihelion nodes. This is now no longer relevant!

    These distances are now shown to be variable, so that the wattages received on Earth, are also variable in accordance with this very same law !

    This is the main reason that climatologists’ computer programmes do not offer up the right answers ?

    1. This is an interesting issue but one that has been thoroughly investigated.

      For a long time now it has been known that planetary orbits are not circular but elliptical. (This got rid of the ‘cycles on cycles’ problem forced on astronomers by the assumption of perfect circular orbits.) In the case of earth this effect is remarkably small; seasons are caused by the tilt of the earth, not its distance from the sun which varies only a small amount. Both effects are easily predicted on the scale of centuries and are not a problem with climate models. Indeed we have some pretty interesting models that relate such subtle effects to ice ages, see Milankovitch cycles: http://en.wikipedia.org/wiki/Milankovitch_cycles

      The problem with climate models is the weird and unpredictable feedback loops earth possesses. For example rising temperatures produce more clouds which reflect sunlight cooling earth, but also replace ice with rock (as the ice melts) which absorbs more sunlight, warming earth further. Which effect is worse? Hard to tell. Sadly climate models are becoming more accurate and they all still agree that temperatures are going to go up and up.

  24. I guess Prof Strassler has given up reading these comments by now… but anyway here goes: I think that the most interesting thing about mass [of course I mean rest mass] is that it isn’t additive: the mass of a system is not equal to the sum of the masses of the constituents. This is not surprising at all — nobody would expect the length of a vector to be equal to the sum of the lengths of two vectors that add up to it [unless they are all parallel] Still, many people have trouble accepting this [eg, the fact that a system consisting of two photons moving in opposite directions has non-zero [rest] mass!], and they invent fictions like “kinetic energy” to make up for the difference. Nobody can stop you from giving names to things, but I think that it is better to really understand things rather than pretend to do so, and so I think people should be taught just to accept that mass isn’t additive— full stop.

    1. One problem is that kinetic energy is a very useful concept that is very much divorced from mass; it is far FAR easier to calculate what will happen when two billiard balls collide using the concept of kinetic energy than to work through the tiny changes in mass (via definition 2) involved. It would be equivalent to demanding people replace the word ‘cold’ with ‘not hot’ (For after all there is no such thing as cold, only a lack of heat.)

        1. If we abandon the notion of kinetic energy and instead use mass being non-additive then mass has to change since we are ignoring energy as a concept. As you state, there are no other options. (You could try and keep other forms of energy but they are about as valid as kinetic energy, if you abandon one it is hard to justify not abandoning energy as a concept entirely.)

          1. Kinetic energy is just one form of energy, for sure. But how it’s stored in objects? One very good explanation and hypothesis is storing kinetic energy in particles forming the object, specially in form of increased spinning frequency of particles. There is no need for mass changes in that hypothesis.

            Besides I have derived so far quite many current physics equations based on my E=mn hypothesis, like de Broglie relation, Compton scattering equations, force equation between objects etc. So far I haven’t had any showstoppers on my route.

            1. I would wonder then how things like potential energy are stored in objects.

              A problem I foresee with the spinning frequency is that two fermions moving at different velocities are still identical, that is they cannot occupy the same space, as you would expect if spinning frequency were of the same kind of property as spin or charge.

    2. It seems to me your speaking about “not to expect the length of a vector to be equal to the sum of the lengths of two vectors” may contain a little mistake. Relativistic mass is (or proportional to) the time component of a four-vector, not the invariant “lenght” of the four-vector. Rest mass is an invariant.

  25. On a nearly totally unrelated note something that I have never quite managed to get my head around is this: if I accelerate any massive particle to a high enough velocity then from my viewpoint it should have enough mass to become a black hole. This of course does not and cannot happen but I have never been able to visualize why. Any ideas?

    1. I will try an answer. Then we will let Matt settle. The singularity point , mass etc have be measured by an observer sitting at the black hole center. Right?

      1. No, black holes can only be observed from the outside, if you get enough mass in one place you will create one This can even be done in theory by colliding two high energy particles together, which is why some people were afraid that the LHC would create black holes and destroy the earth.

        I know that the high speed particle cannot become a BH from its reference frame because to it its mass is constant. But I cannot visualize why it doesn’t in mine.

          1. Yes it is, from its reference frame (that is where it is sitting still and everything else is moving.) unless you believe in a universe where things mass changes for no reason.

            I am not speaking of an accelerating or decelerating particle here but a particle moving at constant, near-light velocity relative to me. By all definitions energy and mass in such an example do not change.

        1. Kudzu,
          OK. You can observe from outside, but the singularity and horizon of no return always refers to the coordinate system of the black hole. So I believe, mass in the equation for the horizon has to refer to the same coordinate system. I think, mass in the black hole radius is the mass according to the first way of defining mass in Matt’s classification.Otherwise one is led to the paradox you are suggesting.

          1. I am not entirely sure what you mean by your assertion, I think you are saying that a black hole’s mass and size, etc only ‘count’ from its point of view, not mine. In which case a fast particle cannot become a BH because from its POV its mass is not high enough. This is in fact why I *know* the particle cannot collapse in the first place.

            My problem then is *why* does the BH’s perspective only matter? This is not true for any other situation I am aware of. I know there must be some trick that is equivalent to what you say, but I don’t know what that trick is, I’m stuck at ‘that’s just the way it is’ at present.

    2. A black hole is defined by a specific mass and a specific radius.If e.g we are talking about an elementary particle(any !) then the required mass and radius (planck’s values ) are attained when its speed becomes c.Then the particle becomes a micro-blackhole(the minimum BH in physics) and according to Hawking,it evaporates (actually explodes) within 10(-40)seconds,emitting a variety of radiation (particles and photons).The interesting thing is that ALL elementary particles attain the SAME maximum relativistic mass when v becomes c—and this is planck’s mass.
      Probably this was a frequent occurrence at the Big Bang.
      It is interesting to think what happens e.g. to a marble sphere when it approaches the speed of light (it apparently disintegrates,because electrons have to hurry up to attain plack’s mass and the baryons stay behind…)

      1. I am sorry but the energy (mass by definition #2) of any particle with rest mass becomes *infinite* as its speed approaches c. All particles will of course reach the Planck mass (in terms of rest mass + energy) on the way to doing this, but it should still be impossible for such a particle to become a black hole because from its reference frame it is sitting still and has no motion energy. (In fact to it YOU would be moving very fast which would mean that from its point of view YOU should become a black hole.)

        This suggests it is not possible to make a black hole just by making a particle move very fast. Colliding two particles yes, but not just by moving one.

        (Also if you move a marble very fast its structure breaks down as the particles in it exceed the binding energy involved in that structure. First the solid breaks down, then its molecules, then the atoms then the nuclei of those atoms (and eventually even into quarks and gluons.) This is not the same as actually splitting the structure which involves applying exactly the binding energy but rather above a certain energy, a molecule say, behave indistinguishably from a collection of free atoms.

        1. A few comments—————
          1.Mass or any other physical quantity NEVER become infinite –ask any physicist if he has ever measured or will ever measure an infinite value for any physical quantity… There are no infinities , nor exactly zero values in physics ,but only in the mathematics used to formulate physical laws–this is a well known problem of physics and many people are trying to correct this situation
          2.When any elementary particle reaches speeds extraordinarily close to c,the relativistic gamma should be corrected such as to prevent mass going to infinity because this is not physical.When this is done properly,the immediate consequence is that relativistic mass of all subatomic particles(namely those obeying quantum laws) grow asymptotically to the same finite maximum value when v=c.If we assume the minimum length to be planck’s length then this max mass is planck’s mass which is the minimum possible black hole in physics
          3.Concerning ME becoming a black hole—-a BH is defined by 2 parameters-mass and radius–you are talking only about the mass which is not enough to define the problem
          4.Concerning the fate of a marble near the speed of light,I still like the idea of seeing a string of fireworks caused by micro-black-hole exposions as the elementary constituents reach the proper conditions at the speed of light
          5.Finally please remember that neutrinos have non-zero rest mass AND travel at the speed of light and nothing weird happens—all of them have (and will always have) finite energies(not so high compared to other high energy cosmic rays)

          1. 1.) A very good point. I should have said ‘tends toward infinity’ but I am not sure how many people understand what that means. ‘becomes infinite as’ I think is a good approximation.

            2.) This is interesting, do you have a reference I could follow? It sounds like very useful physics and may be a reason I do not grasp this problem’s solution.

            3.) Yes mass and radius. However if a particle near c observes you from its reference frame all of your particles will be moving rapidly. If the particle were say an electron moving fast enough to become a black hole then all of *your* electrons would be moving that fast (On average, some slower, some faster.) with their mass and radius the same as you observe for the fast moving electron. Therefore by your logic many of your electrons will have (or exceed) the mass AND have the right radius to become a BH. You yourself would not become a BH but a collection of small BHs and other particles. That you do not do this means that something else besides mere mass and radius is at work here.

            4.) The XKCD related blog ‘What if’ has a post similar to that.

            5.) Indeed neutrinos are a case in point, they must be extremely affected by relativity. I do not know of any upper bound on their energies at present but it would be interesting to explore.

            1. Reference concerning point 2—————

              A.Sternlieb 2013,J.Phys.:Conf.Ser. 437 012010

    3. Relativistic velocity causes particle to lose its energy by creating new particles all the time. You can’t reach the point where there is enough mass to create a BH.

      1. The problem with that explanation is twofold. Firstly a particle does not behave any differently at relativistic velocities; if it is stable when still it will not and indeed cannot create new particles when moving at any speed. This is why cosmic ray protons exist with enough energy to create entire atoms or even billions of atom’s worth of particles. (They can however lose energy by hitting other particles including photons in the microwave background.)

        The second problem is that particle creation is random. In the same way that a mass of radioactive material only approximately follows the half life (especially as only a few atoms remain.) There would be nothing stopping then a particle reaching relativistic speed through sheer chance.

        The thing is, a particle moving at nearly light speed CANNOT become a BH, because from its point of view it does not have the mass. If it could then its experience would be quite strange, it would suddenly randomly become a black hole for no reason and in that case any particle in the universe could do the same thing. My problem is not the creation of a relativistic particle but why, *from my viewpoint* it shouldn’t become a black hole.

          1. Yes, this occurs when *charged* particles are *accelerated* *radially*, requiring all three conditions to be met. Accelerating a charged particle linearly, any sort of uncharged particle or any particle moving at constant velocity are exempt. It is thus trivially easy to come up with examples where no particle creation should apply.

          2. Yes, it’s pretty hard to use neutrons in synchrotron 😉 Anyway, why would particle colliders need so much energy in order to put hadrons near c? Where the all used energy is gone? What causes electron clouds in linear accelerator?

            1. The problem with accelerators is that they are very inefficient. If you want to accelerate a proton (or electron, at these energies it makes little difference) you need to give it a lot of little magnetic ‘kicks’ Each kick involves the manipulation of a large magnet requiring a large amount of energy (since the magnet must be powerful and not permanent – e.g. an electromagnet.) The upshot of this is that only a tiny amount of the energy used is transferred to the particle, the rest is lost as heat.

              Particles can be created (or more usually knocked into) an accelerator beam if there is not a complete vacuum in the accelerator; this is why so many accelerators require a vacuum to avoid the production of ‘mess’ (In the LHC particle production would pollute the beam with everything from muons to Z bosons and more.) the slightest presence of atoms in the beam has the potential to stop said beam entirely.

            1. Yes, linearly. The most common setup is very similar to that of the LHC where charged particles are given magnetic ‘kicks’ by turning on magnets just behind them. In this case however the magnets are arranged in a straight line instead of a circle. This has several disadvantages which explain why radial acceleration is often preferred:

              Firstly the particles cannot ‘loop back’; they get one pass only which means you need a longer and longer accelerator for higher energies. This makes them bulky.

              Since the particle only goes through once it doesn’t get time to build up energy. This restricts linear accelerators to ‘low energy’ applications. (Cathode ray tubes in old TVs are LEs, as are many x-ray machines.)

          3. However,if a particle is unstable (most of them),then their speed may have dramatic effects,e.g,muons reach earth at sea level just because their lifetime (before decaying into electrons)is lengthened 20 times (compared to rest frame lifetime)because of special relativity time dilation effect(their speed is close to the speed of light)

          4. You can use a linear accelerator. They’re less common because instead of running the particles around a loop many times you need a straight tunnel of the required length.

  26. Q,:when people are talking on the universe composition,they say,e.g.(please do not pay attention to accuracy of numbers …):65%dark energy,25%dark matter,5%visible matter(and energy ?),5% neutrinos——a total of 100% which sounds fine.A few questions bother me :
    1.Is it not necessary to be more specific in how those percentages are being calculated,say,by specifying which one is rest mass,or relativistic mass,or trying to bring all components to a common denominator(energy-wise or mass-wise,etc),or being more specific about which frame of reference are the percentages related to(are the percentages being invariant wrt all frames of reference,etc).How is the entire universe “content” being estimated ?Energy-wise,mass-wise ?
    2.I suppose one can raise more similar questions. However I am especially curious about how the neutrino percentage is calculated (energy-wise,mass-wise ?)having in mind what we (think we) know—-that neutrinos move at the speed of light,that their rest mass is estimated to be very small and still decreasing but never zero,etc

    I am sure you understand what bothers me about the nature and validity of the above -mentioned percentages

    Finally I once asked at an informal meeting a Nobel prize winner (in high energy physics) the above kind of questions and after some discussion he admitted not to have an immediate good answer for me

    I would appreciate your opinion

    Thank you so much for your consideration

    1. When these percentages are being mentioned they include (For ‘particle stuff’ such as matter or neutrinos) both the rest mass of the stuff and its motion energy. (So for neutrinos most of that % is in fact their motion energy not their actual mass at rest.) Fortunately aside from neutrinos this makes little difference, the amount of energy in say, a star, due to motion is very small compared to that in its mass. You can thus treat the ‘common denominator’ as energy or #2’s definition of mass and leave it at that.

      It is in fact quite amusing to calculate the exact percentage of the universe’s composition that is due moving normal matter.

      1. Thanks for your answer.
        However when you say “fortunately,,,” I thin it depends on your choice of reference frame.If e.g. you chose a neutrino as a frame of reference (they ALL move at the speed of light—what then ???–motion energy of stars could become enormous with respect to their rest mass…
        I feel I do not yet quite understand the accuracy and validity of those percentages…
        It is important to elucidate this issue because those percentages are being reevaluated and updated all the time according to new findings…

        1. You make a valid point. One of the few things I have trouble visualizing in relativity is that it doesn’t matter what viewpoint we take, the neutrino would consider the universe to have the exact same composition we see it does. Do not ask me how that works out, I know it does but do not understand why. Possibly Professor Strassler or someone more versed in relativity could inform you.

          The interesting thing is that it is easier to ignore this issue’s specifics than to work them out in the same way we don’t need to count all the atoms in a glass of water to know there are 2 hydrogens for each oxygen.

          More specifically the stuff that makes up our universe exerts a pressure, p = wd,: radiation, normal and dark matter exert a positive pressure and dark energy exerts a negative pressure. A collection of still massive particles has a value of w that is nearly 0 (This includes dark matter.). The faster it moves (the more kinetic energy it has compared to rest mass) the higher w gets until it reaches 1/3, the value for massless particles (radiation.)

          When these percentages are calculated what matters most is the pressure and thus w. If all the massive particles in the universe were not moving their w would change slightly and we’d calculate that normal matter makes up less of the universe. The difference would be the amount of kinetic energy currently possessed by the matter in our universe. In a sense the movement of massive particles in these calculations is obscured and the hard problem would be getting an exact or even approximate value for the kinetic energy.

          Take the sun; I can tell from gravity how ‘heavy’ it and all its energy is (Or rather I can tell the total energy of it treating mass as another form of energy.). (If I wanted I could send up a fleet of spacecraft to be more accurate.) But to determine the amount of that which is kinetic energy I would first need to know the average temperature of the sun. To figure this out is even harder; the surface is a few thousand K, this rises and not linearly to millions of K in the core; the density of the sun changes too more predictably as the volume of a given density of material shrinks.

          Taking a whole lot of approximations gives me a value for kinetic energy that is currently less than the margin of error for the sun’s mass that we know. The same is true of the universe as a whole; not only do we have the previous problems but we’re still not sure of things like whether we’ve got all normal matter accounted for (neutrinos were one source we didn’t see coming, so too were intergalactic gas clouds and bodies ejected from star systems\galaxies.)

          In the end you can treat the current figures as quite accurate generalizations unlikely to change too much while the details are complicated and shifting.

  27. By studying QFT I shifted my conceiveing the mass as “inertia” (continuous effect) towards the most mathematical idea of “interaction”, in the sense of “point-like interaction” (point in space-time!). If I push an object I think of its inertia. If I calculate muon-electron scattering I think of amplitude, contraction of tensors, cross section, coupling… and, of course mass, that in my mind is mostly a parameter more than a physical number. The point is: mathematics not only helps me understand and compute, but “she” acquires a very tight correspondence with every physical elementary phaenomenon.

  28. Excuse me! Please pardon me if I didn’t wade through all these comments to see if this was otherwise covered, but doesn’t sense #1 of mass take away the old, wonderful notion of an object’s mass increasing as it approaches light-speed?

    1. Yes, and this is WHY the two definitions are confusing if you don’t pay attention. By both definitions as an object speeds up to c ‘something’ increases to infinity which has some weird effects. For #1 this is called ‘energy’ and kept separate from ‘mas” and for #2 it’s called ‘mass’ (but could more accurately be called ‘total energy of the object’)

      Both definitions produce the same effects when applied correctly but if you mix up your definitions (As I am prone to doing.) you start getting into trouble.

      1. With the first definition the mass doesn’t change with respect to the observer. So even if the object is moving relative to the observer the mass doesn’t change, though the relative energy does.
        With the second definition the mass changes with respect to the observer. So an observer moving quickly sees other things as having higher masses, and it’s the relative velocities of observer and object that partly determine the mass.

  29. I’m a dimensional outlaw; I just equate mass with the force of the earth’s gravity.

  30. Good illumination Professor,
    in reality, there is no “invariant mass”. Because planets move, atoms move, particles move – to avoid Jean-Paul Sartre’s Nausea, there should be “relativistic mass”. But where is constant frame of reference – human observer cannot. So speed of light “c”.
    There is no “mass”, if there is no speed of light. whatever massless is not actually massless, but massless with reference to “c”. The photon is massless because it travel at speed “c”.

    So wherever photos travel in a geometry, mass also travel – I mean along the `hyperbolic geometry of space-time” also – that is gravity ? – the realm of photons or our 3D spacetime ?

  31. Thanks Matt!! One quick question: I’ve seen mass described as the amount of an object’s resistance to a change in velocity. Would this fall into interpretation #2?

    1. The statement is correct if you add a remark: mass measures an object’s resistance to a change in velocity when its velocity is zero or very small compared to c.

      At high speeds, it is (in general) false for BOTH interpretations. See Lev Okun’s article.

  32. Interesting summary. So if one atom of hydrogen with its invariant mass can be called a clock and is moving away from a point of gravity such as the Earth. This clock shall tick faster as its relativistic mass is reducing?? Not because of its isolated RM and velocity but because of b). the distances from Earth are reducing??

    1. ” So if one atom of hydrogen with its invariant mass can be called a clock and is moving away from a point of gravity such as the Earth.” Sorry, this sentence doesn’t parse, so I don’t know what you mean.

    2. No. Any given object has the same rest mass no matter what the gravitational field and time moves faster when there is no gravitational field relative to a gravitational field being present.

      Your hydrogen atom would then in fact speed up as it was moved away from the earth. This may be partly or completely counteracted by the fact that time also slows the faster something moves relative to being stationary. With the right setup you could keep your hydrogen atom ticking away at the same rate relative to a distant stationary observer.

  33. Does this confusion over mass also apply to charge? I would say we can only measure the charge of a body when it’s at rest, and is an invariant whether the body is moving or not.

      1. The minus on the elementary charge of the electron has vexed my imagination. I’m a little biased against minus. I think a teacher once told me it was a coin toss by Millikan as to which sign to assign. Is this mythical? Would everything stay the same if we flipped the sign on the elementary charge?

        1. Physical processes always depend on the product of two objects’ charges. So if you flip the sign of the charge of ***everything*** in the universe, you leave all products of two charges, and therefore all physical processes, unchanged.

          So yes, the choice of -e for the electron is arbitrary; but not the fact that electron and proton have opposite sign charges.

          1. May I add something. The – sign on electron charge goes back to Ben Franklin or whoever made glass rubbed with silk positive and ebonite negative, although they did not know anything about electrons.Once this was chosen it was too late !!!
            Also I understand electrical engineers do not like current going from positive terminal to negative terminal in external wires. But physicists still stick to this old convention. Correct me if these historical facts are wrong!!

        2. The choice of assigning signs to charges was determined by Franklin, however, so Millikan really did not have much of a choice.

          1. Hasirpad,
            Sorry I read your post after writing my post. You are making essentially the same point.

  34. You can live *easily* with the second definition. If I make a hypothesis that E=mn, where E is energy, m is rest mass and n is spinning frequency of a particle then Planck constant is actually the mass of photon. More energy means increased spinning frequency and that mechanism applies to all particles.

    There is plenty of more details with that hypothesis (for example, what is the essence of mass). You can check those details through my home page (if you dare).

    1. Your “spinning” thing is obviously applied to the elementary particles solely because for macroscopic bodies no spinning is implied.

        1. Every day or two, someone new makes a claim on this website, or in an email to me, that his or her theory can resolve most or all of the known anomalies in physics. Well, either every one of these people is wrong, or all but one person are wrong — so your odds are pretty bad. Note also that neither Galileo, Newton or Einstein ever claimed to resolve, in one step, all known anomalies in physics, or even a subfield of physics. So you’re claiming to be greater than any physicist has ever been. NOT LLIKELY.

          1. On a side note, have you considered moderating comments to control for this phenomenon?
            I know many science blogs suffer from it, and I can’t tell specifically what’s so awful about them that would warrant deleting them (and potentially angering some readers), but I really enjoy your answers to sensible concerns on the comment section and it seems like you must get sidetracked very often to discredit outright nonsense (and when you stop answering those claims, on old posts for instance, they thrive to a ridiculous degree).

  35. What is unphysical in bare mass? Why is it needed for defining theory? Is the theory unphysical without it? Who does prevent us from constructing physical theories?

    1. I’m sorry, but these are highly technical questions. This non-technical post is not the place for them. “Bare Mass” is math, not physics; you will never encounter it unless you actually try to calculate something in a quantum field theory, in which case you will need it only as a crutch to make the calculation easier to define.

      1. Does that mean that there is something wrong with calculations (wrong theory) rather than with the mass definition?

          1. No hard calculation needs unphysical concepts. So the problem is with conceptual and mathematical wrongness of the theory, I conclude.

            1. I’m sorry, but that’s a silly remark. Have you heard of Feynman diagrams? Or Green functions? or Gauge Theory? Every one of these involves using something unphysical as a crutch for calculating something physical.

              In fact I don’t know of ANY hard computation done in modern particle physics that doesn’t rely on something unphysical in an intermediate step.

      2. I know everything about Feynman diagrams, Green’s functions, gauge potentials, and even wave functions, but unphysical concept of bare mass is different: its sole purpose is to produce an impression of “absorbing” bad corrections of a good theory. In fact, it is for hiding the practice of discarding bad corrections of a bad theory.

    2. Actually I like your perspective a lot. I’m interested in collaboration and I’m sure you’ll find my paper very interesting. We share pretty much the same perspective.

  36. Hi good point, already addressed by Okun as you said. One can use in the 2nd way the word mass, just as short way of talking, but it may have other consequences,some of which you already mentioned. Buy besides, does the actual mass of a particle/body changes in movement? Does it imply that gravitational effects also are modified creating a singularity as velocity approaches c? Is the mass one should include in propagators changing with movement? There is a whole collection of questions related to the missleading 2nd. use of mass.The definition of mass should be as clear as possible. And, finally, there is a deep and real mistery in that an H atom, for example, does actually weight less than m_e + m_P. Here we have potential energy transformed in mass, for example if we put this atom in a mass spectrometer we measure m_e+m_P-E_binding/c^2 as the actual mass. Here it is unavoidable to include the (negative)potential energy into the mass and it is certainly a generalization of Einstein equation E=mc^2. But this is not the case for energy related to mouvement. Seems to be that this kind of energy, kinetic, is different! Space time is in there….Thanks for your posts and comments.

    1. The correct and clear statement concerning your last point is that “The mass of a system is equal to the total energy of the system, divided by c^2, when the system is stationary.” That’s for interpretation 1 (the one particle physicists use.)

      * To say that a system is stationary is simply to say that its total momentum is zero.

      * Total energy includes the mass-energy, motion-energy [also called “kinetic energy”] and potential-energy [which I call interaction-energy on this website] of the system. In your comment you singled out potential energy. But it is not singled out. The binding energy includes also the motion-energy of the electron and proton. And in the hydrogen atom’s mass, all of its energy is included — again, though, it must be stationary.

      * The difference between interpretation 1 and interpretation 2 is simply that the second interpretation neglects the final condition (that the system must be stationary).

      1. Thanks for your points, I agree. All this stuff is really hidden under the carpet in college and above relativistic physics. In particular the points related mass for compound systems as nuclei and atoms, such as Hydrogen atom. And this is essential for nuclear processes and energy deliver by them. You can read everywhere E=m c^2 but this, in principle, does not include the above, m_H < m_e + m_P. There is something more in the previous statement.

    1. “bare mass” arises as a concept in quantum field theory. It is an unphysical construct needed to define the theory mathematically, but nowhere appears in experiments. As such it is far too technical for this website.

Leave a Reply


Buy The Book

A decay of a Higgs boson, as reconstructed by the CMS experiment at the LHC


The particle physics community is mourning the passing of Peter Higgs, the influential theoretical physicist and 2013 Nobel Prize laureate. Higgs actually wrote very few

POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 04/12/2024

I recently pointed out that there are unfamiliar types of standing waves that violate the rules of the standing waves that we most often encounter

POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 03/25/2024