One of the most challenging aspects of writing a book or blog about the universe (as physicists currently understand it) is that both writer and reader must confront the concept of fields. The problem isn’t that fields are intrinsically that complicated. It’s that they are an unfamiliar abstraction — and novel abstractions of any sort are always difficult both for a writer to describe and for a reader to grasp.
What I’ll do today is give an explanation of fields that is complementary to the one that appears in the book’s chapters 13 and 14. The book’s approach is slow, methodical, and detailed, but today’s will be more of an overview, brief and relatively shallow, and presented in a different order. You will likely come away with many unanswered questions, but the book should help with that. And if the book and today’s post combined are still not enough, you can ask a question in the comments below, or on the book question page.
Negotiating the Abstract and the Concrete
To approach an abstract concept, it’s always good to have concrete examples. The first example of a field of the cosmos that comes to mind, one that most people have heard of and encountered, is the magnetic field. Unfortunately, it’s not all that concrete. For most of us, it’s near-magic: we can see and feel that it somehow makes little metal blocks cluster together or stick to refrigerator doors, but the field itself remains remote from human experience, as it can’t be seen or felt directly.
There are fields, however, that are less magic-seeming and are known to everyone. The most obvious, though it often goes unrecognized, is the “wind field” of the atmosphere. Since we all experience it, and since weather maps often depict it, that’s the field I focused on in the book’s chapter 13. I hoped that by using it as an initial example, it would make the concept of “field” more intuitive for many readers.
But I knew that inevitably, no matter what approach I chose, it wouldn’t work for all readers. (My own father, for instance, has had more trouble making sense of that part of the book than any other.) Knowing this would happen, I’ve planned from the beginning to give alternate explanations here, to offer readers multiple roads into this unfamiliar concept.
Ordinary Fields
In general, I find that the fields of the universe — I’ll call them “cosmic fields”, for short — are not the best starting point. That’s because they are mostly unfamiliar, and are intrinsically confusing and obscure even to physicists.
Instead, I’ll start with fields of ordinary materials, like water, air, rock and iron. We will see there is an interesting analogy between the fields of materials and the fields of the cosmos, one which will give us a great deal of useful intuition.
However, this analogy comes with a strongly-worded caution, warning and disclaimer, because the cosmos has properties that no ordinary material could possibly have. (See chapter 14 for a detailed discussion.) For this reason, please be careful not to take the analogy to firmly to heart. It has many merits, but we will definitely have to let some of it go — and perhaps all of it.
Air and its Fields
So let’s start with a familiar material and its properties: the air that makes up the Earth’s atmosphere, and some of the properties of the air that are recorded by weather stations. As I write this, the weather station at Boston’s Logan Airport is reporting on conditions there, stating that it measures
- Wind W 10 mph
- Pressure 29.71 in (1005.8 mb)
- Humidity 43%
There are similar weather stations scattered about the world that give us information about wind, pressure and humidity at various locations. But we don’t have complete information; obviously we don’t have weather stations at every point in the atmosphere!
Nevertheless, at all times, every point in the atmosphere does in fact have its own wind, pressure, and humidity, even if there’s no weather station there to measure it. Each of these properties of the air is meaningful throughout the atmosphere, varies from place to place, and changes over time.
Now we make our first step into abstraction. We can define the air’s property of pressure, viewed all across the atmosphere, as a field. When we do this, we view the pressure not as something measured at a particular place and time but as if it were measured everywhere in space and time. This makes it into a function of space and time — a function that tells us the pressure at all points in the atmosphere and at all times in Earth’s history. If we define x,y,z to be three coordinates that specify where we are in the three-dimensional atmosphere, and t to be a coordinate that specifies what time it is, then the pressure at that particular place and time can be written as P(x,y,z,t) — a function that takes in a position and a time and outputs the pressure at that position and time.
For instance, consider the point (xB,yB,zB) corresponding to Logan airport, and the time t0 when I was writing this article. According to the weather station whose measurements I reported above, the value of the pressure field at that position and moment, P(xB, yB, zB, t0), was equal to 29.71 inches of mercury (or, equivalently, 1005.8 millibars).
Any one weather station’s report tells us only what the pressure is at a particular location and moment. But if we knew the pressure field perfectly at a moment t — if we had complete knowledge of the function P(x,y,z,t) — we’d know how strong the pressure is everywhere in the atmosphere at that moment.
In a similar way, we can define the “wind field” and the “humidity field” (or “water-vapor density field”) to capture what the wind and humidity are doing across the atmosphere’s entire expanse. Each field’s value at a particular location and time tells us what a measurement of the corresponding property would show at that location and time.
These three fields interact with each other, with other fields, and with external effects (such as sunlight) to create weather. Detailed weather forecasting is only possible because scientists have largely understood how these fields behave and how they affect one another, and have expressed their understanding through math equations that have been programmed into weather forecasting computers.
Air as a Medium
Abstracting even further, we may think of the air of the atmosphere as an example of what one would call an ordinary medium — a largely uniform substance that occupies a wide area for an extended period of time. The water of the oceans is another example of an ordinary medium. Others include the rock of the Earth, the plasma that makes up the Sun, the gas of Jupiter’s atmosphere, a large block of iron or copper or lead, the pure neutron material of a neutron star, and so on.
Each medium has a number of properties, just as air does. Its properties that vary from place to place and change predictably with time can be viewed as fields, in the same way that air pressure and wind can be viewed as fields.
And so we reach a highly abstract level: an ordinary field is
- a property of an ordinary medium . . .
- that can be measured, . . .
- varies from place to place, . . .
- and changes with time in a manner that (at least in principle) is predictable.
Let’s look at a few examples to make this more concrete.
- For the oceans, fields include the current (the flow of the water) and the water pressure.
- The fields of layered sedimentary rock include the rock’s density and the degree to which (and direction in which) its layers have been bent.
- For a block of iron, fields include the iron’s density (the number of atoms in a cube of material divided by the cube’s volume), the orientation of its crystal structure (which might be bent in places), and the average local orientation of its atoms; the latter, usually called the “magnetization field”, determines if the iron will act as a magnet or not.
This manner of thinking is a commonplace, and a powerful one, for physicists who spend their careers studying ordinary materials, such as metals, superconductors, fluids, and so on. Each type of ordinary medium has ordinary fields that characterize it, and these fields interact with each other in ways that are specific to that medium. In some cases, even if we knew nothing about the medium, knowing all its fields and all their interactions with one another might allow us to guess what the medium is.
Cosmic Fields
We can now turn to the cosmos itself. Over the last two centuries, physicists have found that there are quite a few quantities that can be measured everywhere and at all times, that vary from place to place and from moment to moment, and that affect one another. These quantities have also been called “fields”. Just to be clear, I’ll call them “cosmic fields” to distinguish them from the “ordinary fields” that we have just been discussing.
In many ways, cosmic fields resemble ordinary fields. They act in many ways as though the cosmos were a medium, and as though the fields represent some of the properties of that medium.
Empty Space as a Medium
Einstein himself gave us a good reason to think along these lines. In his approach to gravity, known as general relativity, the empty space that pervades the universe should be viewed as a sort of medium. (That includes the space inside of objects, such as our own bodies.) Much as pressure represents a property of air, Einstein’s gravitational field (which generates all gravitational effects) represents a property of space — the degree to which space is bent. We often call that property the “curvature” or “warping” of space.
The list of cosmic fields is extensive, and includes the electromagnetic field and the Higgs field among others. Should we think of each of these fields as representing an additional property of empty space?
Maybe that’s the right way to think about these other cosmic fields. But we must be wary. We don’t yet have any evidence that this is the right viewpoint.
The Fields of Empty Space?
This brings us to the greatest abstraction of all, the one that physicists live with every day.
- Cosmic fields may be properties of the medium that we call “empty space”. Or they may not be.
- Even if they are, though, we have no idea (with the one exception of the gravitational field) what properties they correspond to. Our understanding of empty space is still far too limited.
This tremendous gap in our understanding might seem to leave us completely at sea. But fortunately, physicists have learned how to use measurement and math to make predictions about how the cosmic fields behave despite having no understanding what properties of empty space these fields might represent. Even though we don’t know what the cosmic fields are, we have extensive knowledge and understanding of what they do.
An Analogy Both Useful And Problematic
It may seem comforting, if a bit counterintuitive, to imagine that the universe’s empty space might in fact be a something — a sort of substance, much as air and rock are substances — and that this cosmic substance, like air and rock, has properties that can be viewed as fields. From this perspective, a central goal of modern theoretical physicists who study particles and fields is the following: to figure out what the cosmos is made from and what properties the various fields correspond to.
Imagine that it was your job to start from weather reports that look like this:
- Field A: W 18 mph
- Field B: 29.62 in (1003.0 mb)
- Field C: 63%
and then try to deduce, from a huge number of these reports, what the atmosphere is made from and what properties the fields called “A”, “B” and “C” correspond to. This is akin to what today’s physicists have to do. We have discovered various fields that we can measure and study, and to which we’ve given arbitrary names; and we’d like to infer from their behavior what empty space really is and what its fields actually represent.
This is an interesting way to think about what particle physicists are doing nowadays. But we should be careful not to take it too seriously.
- First, the whole analogy, tempting as it is, might be completely wrong. It may be that the fields of the universe represent something completely different from the ordinary properties of an ordinary medium, and that the seeming similarity of the two is deeply misleading.
- Second, the analogy is definitely wrong in part: we already know that the universe cannot be like an ordinary medium. That’s a long story (explained carefully in the book’s chapter 14), but the bottom line is that empty space has properties that no ordinary medium can possibly have.
Nevertheless, the notion that ordinary media made from ordinary materials have ordinary fields, and that empty space has cosmic fields that bear some rough resemblance to what we see in ordinary media, is useful. The analogy helps us gain some basic intuition for how fields work and for what they might be, even though we have to remain cautious about its flaws, known and unknown. This manner of thinking was useful to Einstein in the research of his later years (even though it led to a dead end), and it also arises naturally in string theory (which may or may not be a dead end.)
Whether, in the long run, this analogy proves more beneficial than misleading is something that only future research will reveal. But for now, I think it can serve experts and non-experts alike, as long as we keep in mind that it cannot be the full story.
31 Responses
Hi Matt Strassler!
“… the cosmos has properties that no ordinary material could have.”
It’s hard not to read that as suggesting the cosmos is a material aether — a dense pool of real particles, just ones with extraordinary properties. Am I reading what you said correctly?
If so, do you mean little oscillators like the Planck-scale ones Scherk and Schwarz proposed in 1974? They were the first to speculate the existence of a mind-bogglingly extreme (~20 orders of magnitude) intensification of Leonard Susskind’s brilliant 1969 insight that non-point particles like protons contain strings.
(When I say 20 orders of magnitude is a bit extreme, what I mean is that if you represent the energy in a proton string as one firecracker, then a Scherk & Schwarz ”super” string would be 60,000 Tsar Bombas, the largest nuclear bomb ever exploded. It’s not surprising that if you fill the entire cosmic vacuum with an unbelievably dense aether of these incredibly energy-intense objects, you wind up with a “vacuum density problem.” 🙂
For the entirely real, non-super strings inside particles like protons, Susskind figured out from the data that the color force that binds quarks behaves like a “rubber band” (his words). That was an especially deep insight since neither the color force nor the quarks that made his rubber bands real were known yet!
As for the “super” string speculation, the energies involved don’t bother me as much as taking the very plausible (and quite real) per-particle 1969 Susskind strings and making them into an aether. All aethers reside in a single inertial frame and thus contradict special relativity. I like special relativity and feel it’s been very well verified.
Another factor is that in 2020, the HAWC Collaboration proved space is way too smooth to be an aether of huge, clunky Planck-scale objects of any variety. Overall, it seems better to go with the data, acknowledge that Einstein was right about special relativity, and discard the experimentally disproven ”super” strings speculation entirely.
The answer to your first question is “no”. It can’t be a material aether made of a dense pool of real particles with extraordinary properties; the medium as a whole would have ordinary properties. [In scientific language: a set of particles would define a preferred rest frame. An example would be the Cosmic Microwave Background, which indeed defines a preferred frame.] Instead, the medium itself must be extraordinary, and I was careful to make clear in the book that I don’t know whether it exists or in what sense it might exist.
For this reason, the answer to your second question and its follow-up discussion is also “no”.
Whatever the structure of the universe is (which I don’t know) it’s more subtle than you are suggesting.
“Much as pressure represents a property of air, Einstein’s gravitational field (which generates all gravitational effects) represents a property of space — the degree to which space is bent. We often call that property the “curvature” or “warping” of space.”
I’m confused by the claim, amplified by the response to Jim Rible, that gravity is unique in being “he one case for which we know the answer” of representing a property of space. As far as I know we have not observed space curvature any more than we have observed the theoretical field line construct of electromagnetic (say) fields. Conversely gravity is to my limited knowledge theoretically as much an effective theory as quantum field theory as they both break down at Planck energies, they have observational limits.
To wit, I believe I have read that reifying curvature has been argued against, firstly by Einstein but also by others I think. Has anything changed that?
This depends what you mean.
I am certainly not reifying anything. What I am doing is telling you what the equations “say”, or more precisely, what they can be interpreted as saying. No such interpretation is unique and I’m not claiming you have to accept it; that’s up to you. You don’t even have to accept that air and pressure exist; space might not, and if it doesn’t, then air (which apparently exists in space) won’t either.
But the fact that you *may* interpret the equations of gravity in this way is different from the situation with the Maxwell equations, where no such interpretation is possible — unless, of course, you are willing to consider something like Kaluza-Klein extra dimensions.
I’m just trying to describe a consistent interpretation of what we know, being careful to state, in the end, that we do not know that this interpretation is unique or even long-term correct. The alternative is complete confusion and discussion of abstractions without any intuition… which makes the subject completely impenetrable by anyone who has not spent years studying the mathematics. I don’t like that alternative. But you are absolutely free to adopt it, as long as you remain consistent with all equations whose validity has been closely tested by experiment.
Hi prof, recently was approved a CERN-Ship detector installation, thousand time more cheap than the FCC. Please, would you explain us on details what is CEARN-Ship detector? thanks
https://arxiv.org/pdf/2112.01487.pdf
PS: we need remember that the CERN-Ship will be made to detect the “Phantom particles”, (or phantom matter) but what is it? thanks
I wouldn’t get too excited about fancy names. They’re just ordinary particles that someone invented a name for. I called them “Hidden Valley particles” in my own work 15 years ago. Ghost/phantom/dark/hidden, it’s exactly the same thing.
If you had to build the SHIP experiment from scratch, not just the detector (the cheapest part) but the full accelerator to power it and the tunnels in which to locate it, you would find it not so much cheaper. Your comparison in costs is like comparing the cost of four tires to the cost of an entire new car; of course it’s a big difference.
SHIP is intended to look for new low-mass particles that are hard to produce, and therefore can only be observed with very high collision rates. This is largely complementary to the strengths of the LHC.
I really enjoyed this article as I always have a hard time understanding fields. All the analogies we usually think of for waves are made of stuff (water, air, etc). But what the heck is a Higgs field? What “stuff” is it made of? Why don’t we just call it the aether (I know—bad connotations)?
However, thinking of the “stuff” as having elements that can be measured, as in your useful and problematic analogy, implies something is going on there. Calling space a vacuum never sounds right to me as there seems to be evidence of something. Maybe not “classical stuff” that involve molecules, but cosmic stuff.
We don’t know detailed answers to these questions. But an aether is a medium, while a field is not. It’s important not to confuse the two; the Higgs field is not, in this sense, an aether. An ordinary field is a property of a substance, so the natural question to ask is: what property of the aether we call “empty space” does the Higgs field represent? (The gravitational field represents the degree to which empty space is warped; that’s the one case for which we know the answer.)
My question might not be the right one, though. Consider it a reasonable but provisional question that might have a clear answer someday, or might end up being unasked.
[This and related issues are discussed in some detail in Chapters 13-14 of the book.]
“The gravitational field represents the degree to which empty space is warped;”
But in textbooks usually it is the “spacetime” that is warped, not space. This is what we are told. Which one is correct?
“spacetime” is more correct. However, there are technical layers to this that I’m glossing over. If you really want the correct answer you have to look at the analogies with electromagnetism and be clear about what is different about the electromagnetic potential vs the electromagnetic field, which gravity generalizes in a complicated way. All of which is to say that both of the statements you asked about are oversimplifications (which is part of why it took Einstein so many years to figure out how it really works.)
We all know the famous quote by Einstein and Minkowski stating that space and time no longer exist as independent entities. Your answer “spacetime is more correct” contradicts this. Either space is replaced by spacetime or not. We cannot have our cake and eat it too. Since in modern physics space has assumed the properties of spacetime, may I suggest that you add space and spacetime in your list of equivocal words in physics? (Your analysis of particle was very helpful to me.)
“Either space is replaced by spacetime or it is not”
No, it simply does not work that way. The goal of physics is to describe what happens to the universe (or a system within it), and also how what happens is seen by an observer. The first crosses time and is appropriately described as a spacetime. That spacetime may be curved. The second requires asking how the spacetime breaks up into spatial slices seen by an observer. Those slices may be curved. The slicing is arbitrary, but useful.
Consider this discussion of the universe as a whole. You will see the words “curvature of space” appear at the bottom of the page. https://people.ast.cam.ac.uk/~pettini/Intro%20Cosmology/Lecture03.pdf
The issue of slicing is regularly discussed by experts. Read just the 2nd half of the first paragraph of https://www.cosmo-ufes.org/uploads/1/3/7/0/13701821/lect.notes-2.pdf
Thanks for those links. But I found the answer to my question in your podcast with Sean Carroll https://www.preposterousuniverse.com/podcast/2024/03/04/267-matt-strassler-on-relativity-fields-and-the-language-of-reality/
In the podcast the word “space” is mentioned 25 times. My understanding is that none of those words refer to “space-like slices of spacetime.” For instance, here you are discussing if space is substance or not https://youtu.be/kCpELmx425w?si=TaM1-QCBIBuheo5G&t=2563. This is a very old philosophical topic, right? It’s predates General Relativity and spacetime and foliation etc.
So in the podcast you are definitely treating space as a thing different than spacetime. All of the words “space” in podcast refer to pre-spacetime notion of philosophical space. At some point Sean Carroll even asks you if you are familiar with Clarke-Leibniz correspondence where they discuss, like you, such things as if space is substance.
Your discussion is also very similar to Einstein’s Foreword to Max Jammer’s book on space where one can find his famous quote that “there is no space without a field”. https://books.google.com.gi/books?id=MdaZa3YXrq4C&lpg=PP1&pg=PA191#v=onepage&q=einstein&f=false I think this is the same space you are discussing.
But the undeniable proof that spacetime never replaced space in physics comes from expanding universe. When physicists say that the universe is expanding they mean “space is expanding.” But if Einstein were true and space was replaced with spacetime they needed to say “spacetime is expanding.” This proves that spacetime in physics never replaced space and space still exists as an independent entity.
So we have thus proved that Einstein was wrong:)
I tried to be very clear that space is a slice of spacetime. They are not the same thing. And yes, in the expanding universe, the space of the universe expands relative to time, where time is defined as the average time variable measured by clocks that measure time using the temperature of the local universe. There is no meaning to say that “spacetime is expanding”; it makes no logical sense. The word “expanding” means “growing over time”, which is something that spacetime cannot do, since time is contained inside it.
However, it is meaningful to say that “spacetime is curved”. It is also meaningful to say “space is curved”, though it is not the same thing. A sphere inside a flat spacetime is curved even though the spacetime is not. A flat plane inside a curved spacetime may be flat even though the spacetime is curved. So when one speaks of curvature, it is important to specify whether one is speaking about the spacetime’s curvature or about the curvature of a spatial slice of that spacetime.
Dear prof, addressing this subject, atoms, magnetism and fields, much is said today about quantum computer and even more AI, both connected to each other. But we must not I ask you please to explain us about it in details, so what is a quantum computer? Even more so with this report below about atoms, magnetism and fields. I hope so you explains us about this new physical technology, briefly in your new text, that is never explained before by physics and physics laws, but only by computers people. Thanks Prof. Regards.
“Physicists Finally Find a Problem That Only Quantum Computers Can Do”
https://www.quantamagazine.org/physicists-finally-find-a-problem-only-quantum-computers-can-do-20240312/
I am not a quantum computer expert, I’m afraid, and do not yet know an exceptionally good way to explain it. I have asked around for suggestions but haven’t gotten a good answer yet. When I do, it will go on my list of things to post about. But it won’t be soon; I have too much to do regarding the topics of the current book.
Based on your knowledge and experience, do you think that it is more likely, or less likely, that we live in a simulated universe? My background is computer science and some things in nature (like entanglement) would be easy to implement in a simulation.
As a physicist, I do not have the ability to put likelihood estimates on things for which there is no experimental or theoretical evidence one way or the other.
I suspect that doing what you suggest is harder than you think, but it wouldn’t influence my opinion; whether or not something is easy to simulate is not evidence that it is being simulated.
“to firmly” => “too firmly”
I had been glad to see you clarify a field as a property of things in your reply to Jason Stanidge. But, part of the problem is that classical “abstract fields” are easily interpreted as if anchored by “points” in the space.
Another issue lies with explanations of physics which attempt to assert “fields as stuff.” This is simply an extension of a class of arguments applied to mathematics called “indispensibility arguments.”
The expression “well-formed” in logic is — more or less — the same as “meaningfulness.” In my own attempts to reconcile your descriptions, this comes into play because you have emphasized that mass is a property of matter. Relative to the mass-equivalence formula, physicists use the expressions “on shell” and “off shell,”
https://en.m.wikipedia.org/wiki/On_shell_and_off_shell
I am seeing this as an (the?) essential characteristic for how your “cosmic fields” are “well-formed properties.” So, cosmic fields appear to be classifiers to be applied when working with matter. This is, at least, how I am trying to understand your views.
A more significant problem for me is that the tendency to interpret classical fields in terms of points in space is not compatible with the uncertainty principle for position and momentum. I had to spend some time reading about position eigenvalues.
There is a tension between logic and physics. The notion of “truth” is a static concept — much like geometry. But, physics tries to explain the experience of a subjective dynamic continuum. It cannot be understood in terms of static inuitions. So, it is as if the differentiation of “position space” and “momentum space” is forced upon us regardless of whether we consider a quantum system or a classical system.
The Wikipedia articles on this topic will eventually get to discussing this in terms of unitary equivalence. One does not find, however, an explanation of why this is significant. Within mathematics, one can discuss operators according to several criteria. It turns out that unitary operators are a particular class of operators which preserve the parallelogram law (My books are ancient, so there may be some refinements of which I am unaware). This is a wider class, for example, than the orthogonal operators related to intuitive notions of rotation.
So, making sense of the laws of motion appear to require treating problems as overlapping “transparencies” (yes, overhead projectors date my age) and these “transparencies” relate to one another by the widest class of mathematical operators which preserve the parallelogram law.
Also, I picked up a translation of Newton this year to sort out some of these questions. It should be unsurprising that the parallelogram law is involved here. In our modern pedagogy, the parallelogram law is taught as algebra. Newton’s “proof” is in the context of physics — it involves counterfactual reasoning in a temporal context. It is certainly not “algebra.”
Logic has words and mathematics has “points.” It is difficult to reason without these things. But, the need to reason with a position space distinct from a momentum space may simply be forced upon us because these “tools” are inadequate for the description of motion.
I am certainly not expert in the remarks I have just made. So, they are not authoritative. They do include some detail, however, which may help you understand the kinds of hurdles faced by your readers.
Thank you for your work. I think your book may be the best I have come across with respect to keeping to the science. Sorry about the length.
I can’t reply to all of this due to its length, but let me note, on this point:
“A more significant problem for me is that the tendency to interpret classical fields in terms of points in space is not compatible with the uncertainty principle for position and momentum. I had to spend some time reading about position eigenvalues.”
Here you are in considerable danger of falling into a trap: mistaking quantum field theory for quantum mechanics. The uncertainty principle in quantum mechanics relates position and momentum operators. No such issue arises for quantum field theory, where the uncertainty principle applies to fields and their rate of change. For quantum fields, positions are just what they are for classical fields: labels of locations, and not subject to uncertainty. Position eigenvalues is a red herring; there is no such thing in quantum field theory, only in quantum mechanics.
This is why classical field theory is a better starting point for understanding quantum field theory than is suggested by your comments. I think some of this may be clearer to you if you go through this series of articles: https://profmattstrassler.com/articles-and-posts/particle-physics-basics/fields-and-their-particles-with-math/ .
Could other fields also be described as bending of space, like gravity, if it weren’t for some particles not reacting to those fields?
Not bending of ordinary three-dimensional space, no; there’s a mathematical theorem that there can only be one gravity-like force, and equivalently that the three obvious dimensions of space can only bend in one sense.
But in a world with extra dimensions of space, then the bending in those directions (or in combinations of known and extra dimensions) would be described by other fields. The most famous example of this was pointed out by Kaluza, followed by Klein, who collectively observed that gravity in a world with a small extra dimension would look to us like gravity plus electromagnetism (plus other objects). [This is touched on in the book (chapters 13-15.)]
That said, it seems clear that not all fields could arise from bending of space, even with extra dimensions. It seems impossible to get everything we observe from pure geometry. That’s fine; space need not be as simple as it looks, and may have internal structure that doesn’t have a spatial interpretation, much as the mere shape of a crystal of iron doesn’t capture its inner capacity for magnetism.
“1005.8 millibarns” I don’t think inches of Mercury are equivalent to (very tiny) areas… (1 barn is 10^(-28) m^2.) I suspect you mean 1005.8 millibars. (Or 1006.1 millibars, which is what my conversion calculation gives me. I wonder if the Logan weather station accounts for some tiny correction…)
For ocean fields, I might also include salinity. But I understand if one does not.
Haha — good catch. And yes, on salinity; similar to humidity in air.
Yes, the confusion for me in the past has been with separating physical fields from abstract, mathematical ones. But it’s definitely helped me hugely to look at the history behind its definition and motives starting with Faraday; who saw the magnetic field as physical and different to the magnets themselves.
Is there a ‘field’ of car speeds on a road system?
Yes, but I say it’s not a physical field in the sense of being physically different to the cars themselves like the magnetic field above. Yet others might argue that it’s physical since the car speeds are physical.
I don’t know what you mean by “physical”. A field is a property of physical things. Do you regard a property as physical?
You are a physical object. Is your height physical? Is your age physical? Is your mass physical? None of these are objects, they are only properties of the objects.
Air is a physical object; is its pressure physical? Pressure is not an object. Still, it can be said to have effects on physical objects. What terminology do you want? We’re now getting into semantics, and I’m loathe to get into a long discussion about it.
An ordinary field is a property of an ordinary medium, and an ordinary medium is an ordinary substance that fills a large region. A substantial amount of traffic on a single highway without exits can be viewed as a one-dimensional medium; it is much like gas flowing in a pipe. In that case, yes: both the density of the traffic and the traffic’s speed — not the speed of individual cars, but the average rate of traffic flow at each point along the highway — can be treated as a field, and will obey equations similar to those of other fields.
The cars are obviously physical objects. Whether you define traffic density or traffic flow as “physical” is up to you; I don’t really care. The important thing is to understand what these fields are and what they do.
[Defining a field of traffic flow on a whole road system is much more subtle. A road system forms a spider-web network, and it does not “fill a large region”. Thus it violates the definition of medium that I have chosen to use. That said, it is possible to define fields on spider-webs (indeed I did it for my Ph.D. thesis), but that takes us to even more abstract realms where I’d rather not try to go today.]
I have to admit that I view properties of things as also physical. A spring is a physical object; the forces within it when compressed are also physical. My height can be measured and confirmed by others using the scientific method; making the measurements of my properties physical and not imagined. It’s an interesting point that I hadn’t thought about until now that: whereas the object is physical, its properties aren’t in the view of others. I’m still thinking about this as I wait for your book to arrive here in the UK.
From your post, I find it obvious to consider space-time as a cosmic field of empty space, analogous to the electron and electromagnetic fields but with its particular properties. And its interaction with the energy and momentum of other fields is analogous to how electron and electromagnetic fields interact with one another. Is my way of thinking about this going in the right direction intended by your post?
Good morning.
The theory of nothing is equivalent to the theory of everything. 🙂
So, simply put, take the center of the Cartesian coordinate system, x=y=z=0, where x=1/Rx, y=1/Ry, and z=1/Rz, i.e. the spacial property of space is a cube bounded by six curved surfaces. So, the intersection of three orthogonal planes can be defined as “empty space”, where there is nothing happening, nil, zero.
As you move away from the intersection and get into cubes bounded by curved surfaces, that’s where thing start to happen and “create” motion, [energy]. So, the “pure energy” is simply the asymmetry of space.
“The temperature of the CMB is a tracer of where matter was in the very early Universe. If the temperature was completely uniform, there would be no seeds for gravitational collapse – no way to form the lumps we see today.” – NASA https://imagine.gsfc.nasa.gov/science/questions/lumpy.html
The CMB is evidence that there was never symmetry, otherwise nothing would exist.