A post for general readers:
Einstein’s relativity. Everybody’s heard of it, many have read about it, a few have learned some of it. Journalists love to write about it. It’s part of our culture; it’s always in the air, and has been for over a century.
Most of what’s in the air, though, is in the form of sound bites, partly true but often misleading. Since Einstein’s view of relativity (even more than Galileo’s earlier one) is inherently confusing, the sound bites turn a maze into a muddled morass.
For example, take the famous quip: “Nothing can go faster than the speed of light.” (The speed of light is denoted “c“, and is better described as the “cosmic speed limit”.) This quip is true, and it is false, because the word “nothing” is ambiguous, and so is the phrase “go faster”.
What essential truth lies behind this sound bite?
Faster Than Light? An Example.
Let’s first see how it can lead us astray.
At the Large Hadron Collider [LHC], the world’s largest particle accelerator, protons collide with other protons. Just before a collision happens, a proton that has been accelerated to 99.999999% of c comes in from the right, and another accelerated to the same speed comes in from the left.
From your perspective, observing these two protons (using fancy electronics, not your eyes) from the LHC control room, the distance between the two protons is decreasing by almost the speed of light from the right and almost the speed of light from the left. So it would seem that the distance between them is closing by almost twice the speed of light — by 1.99999998 c, if you want to be precise. And if the distance between them is decreasing by almost twice the speed of light, then, well, their relative speed (as you see it) is faster than the speed of light.
That must be wrong, somehow, mustn’t it? Otherwise it would violate relativity… right?
But no. It’s not wrong. From your perspective, the two protons are approaching each other at nearly twice the speed of light.
Even simpler: point two laser pointers at one another, and turn them on at the same moment. The light beams will each be traveling at the speed of light, and the distance between them, from your perspective, will be decreasing at twice the speed of light.
When a flash of lightning occurs, light rushes off in all directions. The light moving north moves at c. The light moving south moves at c. From our perspective, standing on the ground, the distance between the light moving north and the light moving south grows at 2c. That’s all there is to it.
But I thought — wait… — huh? — that can’t be ri… — didn’t Einst…? — you’re full of… — I just don’t believe… — … …can’t be true!?! — well I [sound of head exploding.]
This is the kind of problem that sound bites lead to. Let’s take the two-proton example apart, and see what “nothing” and “go faster” actually mean.
The Can’t-Exceed-c Rule
What Einstein’s relativity actually says is this:
- From the perspective of any observer who measures the speeds of physical objects (using a careful laying out of aligned rulers for distance and synchronized clocks for time), no objects will ever be measured to be moving faster than the cosmic speed limit c, also known as “the speed of light in empty space.”
Let’s call this the can’t-exceed-c rule. [This statement of the rule is okay as long as gravity isn’t too important; if gravity matters a lot, then it needs revision… which I’ll return to in a future post.]
The two protons individually don’t violate this rule: we, as observers in the LHC control room, view both of the colliding protons as moving slower than c. And notice: the can’t-exceed-c rule says nothing about the relative speed of two objects as observed by a bystander.
But still, there is a potential threat to the rule lurking here. Suppose we somehow accelerated an observer (let’s name him Peter) to the same speed and direction as the proton coming from the left. Peter would then move along with that proton, and would view it as stationary. From our perspective, as illustrated in Figure 1, we might think Peter would then view the proton from the right as approaching at nearly twice the speed of light. That would violate the can’t-exceed-c rule.
Similarly, if Paula were traveling with the proton coming from the right, we might think she’d view the proton from the left as moving at nearly twice c.
Here’s where relativity of space and time as Einstein intuited it, and as experiments confirm, steps in to save the can’t-exceed-c rule. The point is this: because Peter is in motion relative to us, Peter’s view of space and time is not the same as ours. This is key. Because of his differing views, Peter will lay out clocks and rulers differently from how we, sitting in the LHC control room, would do so. Therefore, the way that Peter measures speed — the distance covered by a moving object over a certain amount of time — is different from how we are doing so.
And that’s why, even though we measure the two protons approaching each other at nearly twice light speed, Peter will measure the right-moving proton approaching him below light speed. (The analogous statement would be true for Paula, traveling along with the other proton.)
In summary: from the perspective of an observer traveling with either proton,
- we and the LHC control room are approaching at a speed of 0.99999999 c
- the other proton is approaching at a speed of 0.9999999999999998 c .
Meanwhile, from our perspective
- Both protons are approaching us (from opposite directions) at a speed of 0.99999999 c
Different observers simply disagree. Yet all are simultaneously correct. This is characteristic of things that are relative (i.e. perspective-dependent.) [You and I think the Sun is bright, but an observer out at Pluto would think the Sun is dim; there’s no logical contradiction, and all of us are correct.] The difference, again, stems from different (but equally valid) ways of measuring distances and durations.
[One thing we do have to agree on: if A views B as approaching with speed v, then B must also view A as approaching with speed v. Our perspective and those of Peter and Paula are indeed all consistent with this requirement.]
Notice that no one’s perspective violates the can’t-exceed-c rule. The universe preserves this rule, as described by the math of relativity, no matter how you try to trick it into an exception. Yet that math doesn’t promise or assure that this rule applies to the “relative speed” of two objects — if “relative speed” is defined as the rate of change of the distance between two objects as measured by an bystander.
As always, though, definitions matter. If instead we defined the “relative speed” of two objects to be the speed of the second object as measured by an observer traveling with the first, or vice versa, independent of any bystander, then indeed that “relative speed” cannot exceed c. This is the definition implicitly used by physicists most of the time. But as with all definitions, it’s an arbitrary choice. It’s not the intuitive one most non-physicists would use.
From Sound Bite to Understanding
So what does “nothing can go faster than the speed of light” really mean? “Nothing” means “No physical object”, and “Go Faster…” means “can be measured by an observer to be moving…” And altogether the sound bite means,
- “no physical object, from the perspective of any particular observer, can ever be measured to be moving faster than the speed of light in empty space, c.”
Even this comes with additional fine print. (The observer should be inertial and must make the measurement using an inertial frame of reference, and gravity had better be irrelevant; and I’ve left out some details about what “observer” really means and how the measurement is really made.) I’ll return to some of that fine print another time. But I hope it’s clearer now what the initial glib statement doesn’t mean.
Fundamentally, a sound bite or TL;DR approach to science can’t ever work. Many of the remarkable and non-intuitive features of the universe are within the grasp of non-scientists, but they require more than a single sentence, or even a paragraph. What sound bites do in relativity is similar to what they do in politics: they make us think we understand something, while actually obstructing the path to knowledge.
36 Responses
The claiming that relativity “sets aside the concept of energy” might not correct. As a static in extra space, the expansion is represented by “Red shift”, but in order to keep the speed (no expansion), the #Energy (frequency) is increased. ?
Pythagorean relativistic mechanics:
That the Black body radiation possesses inertia was first pointed out by F. Hasenöhrl.” — Max Planck, 1909.
Radiation in a moving cavity. This case is of historic interest, since it can be treated by electrodynamics alone, even without relativity theory. Then one necessarily comes to ascribe momentum and thus inertial mass to the moving radiation energy. It’s interesting that this result was already found by F. Hasenöhrl before the introduction of relativity theory. However, his conclusions were in some points in need of correction. A complete solution of this problem was first given by K. v. Mosengeil.” — Wolfgang Pauli, 1921.
ADDENDUM: Equations for age gradient and moving-frame light velocities.
For all equations below, forward velocity is the z+ direction, xy is the transverse plane, and backward is z-.
Age gradient (or “timestamp slope”) using moving-frame z length (c-limited):
a_g=-v/c^2
Relativistic age gradient across Lorentz-compressed rest z length (can go to infinity):
a_γ = γ * a_g = -v / [c * sqrt(c^2 – v^2)]
Forward light-path length ratio R (moving-path over rest-path):
R = sqrt((c+v)/(c-v))
Light velocity (positive) at angle theta from forward velocity z+:
c_θ = c * R^(-cos θ) = c * [sqrt((c+v)/(c-v))]^(-cos θ)
Reciprocal light velocity (negative) at angle theta. Positive-negative product is c^2:
c_θ = c * R^(cos θ) = c * [sqrt((c+v)/(c-v))]^(cos θ)
Forward (z+) light velocity:
c_z+ = c_(θ=0) = c * sqrt((c-v)/(c+v))
Transverse plane (xy) light velocity:
c_xy = c_(θ=π/2) = c
Backward (reverse, z-) light velocity:
c_z- = c_(θ=π) = c * sqrt((c+v)/(c-v))
Matt Strassler wrote (September 1, 2022):
> […] the speed of the second object as measured by an observer traveling with the first, or vice versa, independent of any bystander […] This is the definition implicitly used by physicists most of the time.
Indeed.
In contrast, “the magnitude of the Galileian difference of the velocities of the two objects, as measured by a selected bystander” can be generally quite different from the (proper mutual) speed of the two objects wrt. each other; as well as generally quite different for different(ly selected) bystanders;
except as far as the separate magnitudes of both velocities are both much smaller than “c”.
> But as with all definitions, it’s an arbitrary choice.
Since the definition of the one notion, or quantity, involves the selection, case by case, of a particular (and suitable) bystander, and the definition of the other does not, the latter is this much simpler, and in the jargon of physicists: “proper”.
Matt Strassler wrote (September 3, 2022 at 6:25 PM):
> […] All motion in our universe is relative. […]
At least: “speed” is widely understood as “relative”; namely as being assertable “of one, in reference to another (suitable) one (with its suitable reference system)”.
That’s a reason for not burdening the word “relative” with the task of distinguishing the names of differently defined notions of “speed”;
i.e. for dismissing the suggested terminology “relative speed” in favor of “magnitude of Galileian velocity difference”, for instance.
Erratum: Delete next-to-last para, “The weird … sloppy, though.” (Grammarly is great, but it can’t catch everything.)
Hi Matt Strassler: I don’t think I can answer your most recent questions without explaining more precisely what I mean by “age gradients” since, at this point, I suspect there is no existing terminology for this idea after all. As usual, please feel free to truncate or delete this long answer as you see fit. Just so you know, I will post a copy of the content below on my website, regardless of whether you keep or delete it here.
———-
Age gradients, a_g=-v/c^2, apply at all SR scales, including classical, so let’s forget particles for the moment and talk about human-scale clocks.
Picture two clocks, A and B, 100 m apart on the z-axis and linked by lasers [1]. What must one do to move this dual-clock system into the v_z = 0.6 c frame?
(1) Move B, the right-hand clock, 25 m left to create a new separation of 75 m. This movement — others are equally viable — presets the clocks to the 0.75 Lorentz factor for 0.6 c. Physical compression is required since each clock accelerates separately. If you don’t pre-compress, you get “frame ripping” along the z-axis (Bell’s string paradox).
(2) Reset the time on B by -200 ns. Why? Because the instant B starts moving at 0.6 c, light from A will take longer to reach it. Once accelerated, the B clock needs an extra 200 ns of “slack time” to let the light catch up. Without this spare time, it can no longer stay synch with A in the new frame.
(3) Wait if you want to. Contrary to what folks tend to assume, boosting is not mathematical magic. It’s an objective process with multiple steps, and you can do those steps in different ways with slightly different outcomes. I could, for example, have added 200 ns to A to create a different, but still fully SR-compliant, Lorentz transformation of the original system. The acceleration of complex objects is quite a messy process.
(4) Accelerate both clocks “instantly” to 0.6 c in the z+ direction. It is only then that the boost abstraction — the mathematical abstraction that there is a self-existent “space” for understanding how the system behaves — even kicks in. Without prep steps 1 (Lorentz contraction) and 2 (application of the age -v/c^2 age gradient), you don’t get a _self-consistent_ Lorentz transformation of the original system.
The age gradient — the smooth -v/c^2 gradient one must add to transform a system correctly from one frame to another — is a separate feature of frame transformation. Just as you must pre-compress length to prevent “space ripping” (Bell’s string paradox), you must also pre-adjust clock settings to avoid “time ripping” of the object in the new frame, that is, to keep it in sync with itself. Ensuring time dilation is a _dynamic_ effect of a different sort. That’s why you can, if you wish, insert the “wait a bit” step 3 after finishing steps 1 and 2. In sharp contrast, if you don’t pair and correlate the Lorentz contraction and “age-sloping” transformations, you end up with an inconsistent result.
Age gradients are part of converting twin-type paradoxes into fixed, non-relative rates within the most recent launch frame that encompasses all of the interacting structures. There’s also a launch-frame energy hierarchy.
Does any of this affect quantum, though?
Yes. In particular, the complex-plane phases of Schrodinger momentum wave functions are age gradients. Rest-mass quantum frequencies acquire age gradients in the direction of motion, twisting their amplitudes into a helix.
I’m sorry if I’m introducing new physics in any or all of this. I was sincere when I said I’m just trying to figure out the names of some of this stuff. When I find something interesting — Glashow’s cube is an example — I assume it’s well-known, and I don’t yet know the right keywords to look it up.
But if there are no existing keywords, then maybe it is new. But why would such an almost absurdly simple concept get overlooked for over a century?
Increasingly, I suspect it’s because the Poincare symmetries work so well that the instant we say “object,” our brains switch over to moving-frame space and time. Why not? They are, after all, absolutely valid coordinates. Thus the fact that we changed frames doesn’t even occur to us.
The danger of that seductive little mental flips is that it undermines how fiercely SR applies in every situation. SR applies even ones when it converts ordinary into weirdly sloped time entities whose dynamics are better modeled by making the speed of light reciprocal in the forward (c/R) and backward (Rc) directions. (I know there are papers on that, but they use 1/2 c and infinity, and that’s just wrong; space is always relative.)
The weird and very non-real. That’s wrong, but it makes it tempting to mentally switch back to the moving frame instead of entirely forcing the systems into the rest frame. That’s a bit sloppy, though.
———-
[1] The two-clock system details are: Both clocks use Doppler to stay at rest relative to the other. Both use 667 ns (200 m / c) send-return signal loops to ensure 100 m separation. Both send their current times in each send-return loop, and both synchronize their clocks based on the other clock being 333.6 ns distant.
I mentioned a number of quantities in the above comment, all of which have precise definitions. I’m not sure whether Matt Strassler’s new spam filter will accept text-form equations, since they resemble code and could easily trigger a generic exclusion in an untrained filter. I’ll give it a try after this test comment, though most likely, that would also get me banned from posting.
I’ve never heard of someone contemplating the relative speeds in relation to the cosmic speed limit. (But maybe I’m too stuck with the relativistic physics.) So this was a hard example to retcon the intended wrong formulation out of.
Sure, it reminds of astronomical observations of angular speeds, where you have to be careful since appearances deceive. (Or in terms of the example, rotate a laser pointer and observe the spot “speed” on a distant wall exceed the cosmic speed limit.)
The issues of astronomical observations of angular speeds is a little different, but equally important, since it has to do not with *measured speeds* but with *apparent speeds* (where light travel time has not been accounted for.) Another case I plan to write about.
As for today’s case: I know I’ve been asked about this puzzle at least a dozen times by non-experts, so it must be a pretty common confusion.
Another great write up. If I could ask a question, with a slight modification to your scenario. I’m in my space ship, initially at rest relative to a distant star. I see the light from that star coming to me at the speed of light, and a certain “color” (frequency). I then accelerate to .999999 the speed of light, relative to that star. I would still see the light from that star approaching me at C, but wouldn’t it be frequency shifted? If so, as the light shifts frequency, as my velocity relative to the star was increasing, wouldn’t the frequency eventually shift into the ultraviolet and I wouldn’t be able to see the star anymore? Wouldn’t those high energy UV photons (or higher) start doing damage to me and my ship?
Yes, the light would be blueshifted, would become invisible to your eyes (though not a UV light detector), and would become dangerous. Effects like this are widespread in cosmic ray physics, where the collision of a ultra-high-energy cosmic ray with some poor unsuspecting slow particle can create a dramatic effect; examples are cosmic ray proton (say, 1 PeV) + microwave photon from the cosmic microwave background (0.001 eV) can lead to proton + newly created electron + positron, which a microwave photon normally couldn’t do. There are lots of other examples, such as https://cerncourier.com/a/cosmic-ray-particles-could-give-information-about-neutrino-mass/ .
Incidentally the reason James Webb telescope aims at infrared light is that starlight from distant galaxies has been redshifted (by the universe’s expansion) to the point that our eyes can’t see it. It’s a real effect.
I don’t understand. The whole article is about relative motion and measuring relative motion. Two cars moving in opposite directions one with 30 km, the other with 70 km, will measure 100 km relative to each other. This does not mean that either of the car is moving with 100 km. Why would this be any different for the speed of light.
“This does not mean that either of the car is moving with 100 km. ” Yes it does. ***All motion in our universe is relative.*** If someone in the first car measures the second car to be moving at 100km/hr, then yes, that’s its speed (in the reference frame of the first car, which is just as good and just as bad as the reference frame centered on the ground.)
From the Sun’s point of view, both cars are moving at several km/second. Are you saying the Sun is wrong?
Matt, have you ever browsed the Google newsgroup sci.physics.relativity?
I ask because relative velocity, closing velocity etc is what many skeptics there believe contradicts special relativity, and your post gets to the heart of their confusion precisely and quickly IMO. There’s also the experimental evidence for relativistic kinetic energy which even the most hardened skeptic finds difficult to refute: The velocity of an object approaches a limiting c, yet the temperature of the stationary target it collides with continues to increase unbounded as shown by experiments around the world.
“Peter will lay out clocks and rulers differently from how we, sitting in the LHC control room, would do so”. Yes, that’s the crucial point for me or equivalently: “Peter will label the same event point using his clocks and rulers differently…”
In every day life we can see this in the case of the labeling of space coordinates and their corresponding differences by comparing how a driver in a car will do this compared to someone on a road using their rulers. It’s something we take for granted. Then Einstein came along and claimed this was symmetrically also true for the labeling of time coordinates for events in a way that maintained a constant universal speed limit c for all observers.
No, I haven’t. Experimental evidence for relativistic kinetic energy is also completely obvious at the Large Hadron Collider, which wouldn’t work without it; the whole principle of the the machine is based on it, and it all has to be timed with extreme precision, requiring precise knowledge of the speeds involved. The protons move at nearly the speed of light, but bunches collide over a tiny fraction of a nanosecond, and do so every 25 nanoseconds, while traveling around a tunnel 27 km (i.e. 90 microseconds) long, and they do so for hours. There’s no wiggle room to mess with basic relativity in such a system — so anyone who doesn’t understand this is spouting silliness.
“No matter how fast the light is” (Pendapat Saya 🙂
spontaneous is faster than light…
Lorentz transformations work because they project four-dimensional structures into an observer’s xyz space. But there’s more structure to such projections.
Would you happen to know the standard name for the age gradient — my phrase, sorry — that accompanies Lorentz compression?
It maxes out at sbout -3.34 ns/m and makes the front end of your LHC proton 2.78 × 10^(-24) seconds younger than the back end.
Also, is there a name for the relativistic version of this age gradient, which for high-gamma objects is gamma*(-3.34 ns/m)?
I’ve tried to look this up but had no luck.
You are implicitly assuming something about how you are measuring age; how was the proton born?
All you are really doing is measuring the time it takes for light to travel from one side of the proton to the other.
If you image a proton at rest you see quark positions guided by conservation rules. If you image the same proton at 0.99 c, you see its trailing quarks in slightly retarded positions relative to the expected pattern.
Watching a sequence of such images would be like watching a movie where the left side of each frame is a half second older than the right side. How the proton was created should not matter.
As you aptly noted, lightspeed delays are the root cause of retarded quark positions. But if Lorentz contraction is experimentally real and measurable, then trailing-side quark-position retardation should be just as real, since they are part of the same 4D-to-3D projection package.
Erratum: _Advanced,_ not retarded. Rest views clocks at the trailing edge of a moving object as _ahead_ of clocks in its leading edge. The age gradient a_g = -v / c^2. Apologies for any confusion. (I should read my own papers.)
I’m still not clear what your question is. Are you asking about wave functions of objects in motion? We don’t need to worry about something as complicated as a proton. Why don’t you think carefully about the properties of a boosted hydrogen atom or positronium (but be careful, because you need to account for virtual photon emission from the electron, as well as virtual electron-positron pairs; you are treating it as a relativistic system, after all.)
Please see the main thread for my (long) response.
Or more simply: no it is not true, things can move faster than light in a medium.
While this is a true statement, do you really think it’s more important than saying, “yes, it is true, as long as you mean ‘the speed of light in empty space, c.’ ” ? That is, is “no” really more profound, and more insightful, than “yes” with proper definitions? I don’t buy it. There’s almost nothing more important in physics than local causality, which is the formal way of saying “yes”.
It is possible to travel faster than the speed of light, but to explain how, I need to introduce some basic field structuring and fourth vector structuring concepts. [Extended nonsense removed by host.]
Cherenkov radiation came to mind with the title on this one.
Regardless, I think the point of Relativity is lost when comparing “speeds” based in simultaneity from one’s own perspective. Events separated in space are not simultaneous in different frames, and an observer of two separated events is simply subject to the interpretations of what can be observed from her own frame… in this case, a near twice light-speed rate of closure from a central perspective of one comparing information arriving from two disparate light cones.
What is preserved from any perspective is *causality* (and retro-causality?). As you point out, Relativity implies that the way any observer measures both time and distance from their own frame will always insure that no one breaks the rules… but fundamentally such that from any perspective the protons reach the same point and collide *before* resulting in a spectacular spray of particles.
I think the more interesting question is what this means when we venture to follow the light cones backward as far as we can?
You’re thinking like a physicist. You know too much. Cherenkov radiation is the least of the conceptual problems with relativity, if you’re not a physicist.
As you say, a better way to view relativity is through causality, whose structure arises at space-time points. In the absence of gravity (and thus space-time curvature) we can however, extend our understanding from a single point across all of spacetime, if we have a consistent frame of reference. I’m not sure what you mean about comparing speeds “based in simultaneity from one’s own perspective.” In the absence of gravity, speeds of objects at different locations can be compared… and this is true in virtually all particle and atomic and molecular processes on Earth, since gravitational effects are very, very slow.
Is there another caveat needed for things that can reduce the vacuum index of refraction to slightly less than 1, thus slightly increasing the speed of light moving between two Casimir plates to a bit more than c? Of course, this effect would be so ridiculously tiny that I don’t see how it could ever be measured.
I don’t know if “caveat” is the right word so much as a “caution” to again read the fine print very, very carefully. Relativity constrains physical objects and information. Index of refraction less than one affects parts of waves, but still doesn’t allow photons — objects — to exceed c. In https://en.wikipedia.org/wiki/Refractive_index there’s some discussion of refractive index below unity.
In the LHC, lead nuclei Lev and Dex approach each at equal and opposite near-c velocities. They appear in the CMS detector as Lorentz-compressed oblate spheres with similar thicknesses that merge to form a single excited oblate sphere in the CMS rest frame.
The rightmost nucleons of Lev and leftmost nucleons of Dex collide first, initiating the formation of quark-gluon plasma in the CMS frame… [Abridged by editor since it had an error and was quite long.]
So, three questions:
(1) Which of these historically incompatible models is correct?
(2) Which one does CERN use?
(3) Why?
I think this is more complicated than necessary. You can ask the same thing of proton-proton collisions: how do you describe the collision in the lab frame, in one proton frame, or in the other proton frame? You can do any one of the three, it’s just a translation of the physics from one perspective to another, if you do it correctly. But the easiest one for LHC-relevant phenomena is the lab frame in which both protons are flattened into pancakes by length contraction. The same is true for heavy ion collisions. So (1) all three are correct, they are not historically incompatible — cause and effect work perfectly in all descriptions, thanks to relativity; (2) scientists use mostly the lab frame where both nuclei are pancakes, and (3) they do so because experience shows this is easier, partly because protons and neutrons are simpler when fast-moving than when at rest, but not because there’s
Yep! The bogus part of my thought experiment was simple in retrospect: In the nuclei views, I kept the plasma moving at the _same_ velocity as that of the other nucleus creating it. It doesn’t, of course. It slows down just enough to match the CMS frame view of a plasma at rest.
Had I done the transforms _before_ jumping to imagery, the deceleration of the plasma would have showed up trivially. All I did was build a bogus and deceptive visual image. Wow!
Again, thanks.
Speed is the distance covered in Space at a period, if it is SpaceTime, Space itself adjust to the time.
If it is more Space light has to travel more speed to Cover the speed “c”… or as a paradox, create a curve, that what we call the “mass”?
The claiming that relativity “sets aside the concept of energy” might not correct. As a static in extra space, the expansion is represented by “Red shift”, but in order to keep the speed (no expansion), the #Energy (frequency) is increased.
Without ‘relativity’ we cannot get the “Moon and Sun” at the same size (even by excluding the Parallax libration). This “Time dilation” is the physical reality, where ironically we get the ‘Rest mass’, not a physical reality but only a Mathematical “zeta value”.
Riemann laid the foundations of the mathematics of general relativity.
Srinivasa Ramanujan under Godfrey H. Hardy
worked out the Riemann series, the elliptic integrals, hypergeometric series, the functional equations of the zeta function, and his own theory of divergent series, in which he found a value for the sum of such series using a technique he invented that came to be called Ramanujan summation.
“The study of Euler’s works will remain the best school for the different fields of mathematics and nothing else can replace it.”- Carl Friedrich Gauss.
The existence of “Closed forms for polynomial roots (probability density)” hinted “at lowest Energy the Energy Limit itself as a Equation to Matter (not the Mass at first)”, equates Space & Time to SpaceTime.
Linear Regression analysis was conducted to establish the correlation between global and cartesian planimetric coordinates of and regression models generate results in GPS (Geodesy).
The thought of St. Augustine of Hippo revolved around dualisms, inheritance of Plato and Oriental Manicheans, brought the belief Systems to #Holographic (2D), like Pythagorean and Cartesian coordinate system by Soma or Surā.
But in egyptian Amun or Amen (truth?) , god of the air (2D phonon or Soma), the left side has Female symbol ♀️ and the cone head tilted to female side, depict the anti-gravity tree growth pulled to “lowest #Energy”, called the Mass.
Figure 2 confuses me. It would seem to be a totally different case than Figure 1. With Figure 1 the left proton moves at 0.9999999c and Figure 2 showing 0.999999999998c. I would think an observer accelerated to the same speed as the traveling proton would view it a stationary and the the article says so. “…would view it as stationary” Then the 0.9999999998 shows up in Figure 2.
I took it to mean the Figure 1 and 2 were the same case from a different perspective but I guess only “consistent” was meant. “That this picture can be consistent with Figure 1”
Or maybe I am missing something here.
Hmm — Are you confusing “from the left” with “to the left”? I’m trying to see where I might have flipped one with the other in my text, but can’t find an error. The statement was that Peter (drawn on the left of Figure 2) is stationary with respect to the proton coming *from the left*, and sees the proton coming *from the right* as moving at 0.999999999998c .
If there’s something inconsistent in my text let me know, but as far as I can see you’ve flipped left and right in the situation I’ve described in the text.