A moment of levity…
The post got long.
Summary: GR’s “curvature is caused by ‘stuff’” is a good bit less arbitrary that it may seem.
Part A and Part B below can be read mostly separately. Part B is a lighter read.
Part A
While it’s true that another layer of “Why?” or “How?” can always be added in, some layers feel a lot closer to bedrock than others.
Physicists are happiest when physical laws can be connected to a (subjectively unobjectionable) symmetry. And when they can’t, there is likely active work to dig deeper.
A familiar example would be conservation of momentum. Why should momentum be conserved in physics? It turns out to be directly derivable from the notion that the laws of physics are unchanged when going from one point to another. That deeper principle is fairly unobjectionable. You could still ask “Why should the laws of physics be the same everywhere?” and that would be a valid question, but also… Why shouldn’t they? This “symmetry” of the laws across space seems like a fine starting principle (that also happens to be consistent with observations).
Should the laws of physics be the same right now as they will be a few minutes later, or as they were a few minutes ago? If that principle of “time translation symmetry” is unobjectionable, then you automatically get energy conservation. That is, energy conservation isn’t something stuck into physics arbitrarily.
Should the laws of physics be the same when I look this way versus that way? If so, then you can prove the conservation of angular momentum. You don’t have to add it. It’s there for free if you accept rotational symmetry.
Jumping to relativity: should the laws of physics be the same for all observers, regardless of their relative velocity? Seems like a good thing to require. This plus the injection of a very simple way to calculate distance in spacetime gives you all of special relativity.
All of the above gets packaged mathematically into something called Poincaré symmetry. So the questions of “Why conservation laws, and why special relativity?” are answered by “There is an elegant symmetry group in the universe that, if present, implies all of those things.”
Going further: General covariance says that the laws of physics shouldn’t depend on the details of your coordinate system. This opens up the possibility for curved geometries. GR starts to creep in now.
Another principle that physicists like is “If nothing forbids a thing, that thing should happen.” If physics should be fine with curved spacetime, then spacetime should be able to curve. What should decide the curvature? How about the curvature of spacetime is tied to precisely the things that are consequences of the symmetries of spacetime? Energy and momentum aren’t arbitrary choices; they are directly tied to spacetime symmetries. So, let’s package those into the only fully symmetry- and covariance-respecting form you can write down. What curvatures might this form relate to? Well, there’s actually only one tensor you can write down that obeys all sensible requirements.
So, at a bird’s-eye view, physics should be fine with curvature, and rather than introduce something out of left field to dictate that curvature, let’s tie the curvature (indirectly) to the symmetries of spacetime, i.e., to the sources that are preserved (conserved) by the symmetries.
Part B
A more advanced – but much more elegant – way of getting to all this is to invoke the action principle, which underlies literally all of physics and says that a physical system evolves in whatever way minimizes a quantity called the “action”. If you imagine a curve-able spacetime that respects general covariance and also has some stuff in it, and you write down the simplest action you can – and it’s really quite simple in form – then out pops general relativity. It sort of has to be.
Can you still poke a stick at this layer? Poke away. Maybe general covariance is faulty. Maybe the principle of least action doesn’t apply here. But, those are much less arbitrary foundational principles for GR than what may be implied in a typical lay description (where “stuff” magically curves spacetime).
Oh, it is worse than that. Our most ‘comprehensive’ theory of physics, the ‘Standard Model of Particle Physics’, is ferociously complex and breaks down more readily than a British sports car on a slightly rainy day. By comparison, the Einstein field equations of General Relativity are quite well behaved and helpfully predict a lot of phenomena…as long as you stay away from places where spacetime gets knotted.
It should be understood that ‘forces’ as we observe (or rather, infer) them from everyday experience are really abstractions of more fundamental mechanisms going on behind the veil of the macroscopic world, and there is a limit to how deeply we can ever really understand them based upon our intuition and mental models of the world as we personally experience it. But the o.p asks an even more fundamental question, to wit:
@Pasta reference the ‘action principle’ which provides a kind of general approach to describing what will happen to a system and how it will evolve given a distribution of energy and mass from which a specific dynamical law or principle that can be used to describe a particular class of phenomena but not why some actions are required and others are prohibited. For that, you may need to go to something like constructor theory, which may be the route to a more fundamental theory of physics or might just be complete bunk.
Stranger
I don’t think my comment was clear. What I mean is that the external gravity field being applied to make the masses curve spacetime muddies the analogy. It interferes conceptually with the concept that the matter causes the curvature itself.
I’m well aware that analogies and models are not the end-all-be-all explanation. Rather, they serve a limited purpose to demonstrate a single concept. The problem comes when people fail to grasp the limitations and try to extend the analogy beyond its limits.
@Pasta , I do appreciate your long post.
But to a layperson (like myself) it can be extremely eye-opening. When I was first told that gravity was not a force, but a curvature of space, intuitively it made no sense. But after seeing that demonstration I was able to visualize how a moving object could ‘curve’ but be going in a straight line across a warped space. And despite soon enough realizing that the demonstration used gravity to simulate itself, I still feel it is useful to demonstrate the concept of warped space - if not to physics students at least to the everyman.
It’s a very separate discussion, but, yeah, I have some… concerns. It presses all the right buttons to generate some level of media stir, and so it has done at least that.
Well of course gravity exchanges kinetic energy and momentum between masses– that’s practically the textbook definition of a force. When we say that gravity is not a force, what we really mean is that gravity is (so far) best explained as a geometric theory not a particle-exchange theory, whereas for the other three forces it’s the other way around. Probably a “theory of everything” would have to show that particle-exchange models and geometric models are ultimately inter-equivalent.
To be fair, it isn’t really possible to make any kind of analogy to a theory of gravity without somehow invoking gravity; it is the only ‘force’ which we obviously and intuition observe in action in an everyday context. All other fundamental ‘forces’ occur at a scale that we can only infer their action.
I’ve followed developments in constructor theory casually for about the last twenty-odd years since I read Deutch’s book, and have ended up putting him in the same bucket as Penrose and his speculations on twistor theory in The Road to Reality of not being able to decide whether it is brilliantly inspired or complete nonsense. It is at least a novel approach that doesn’t have the limitations of just trying to tweak existing interpretations of cosmology and quantum mechanics, but of course that doesn’t make it a valid approach.
Stranger
This is why I welcomed corrections, and I ask this question sincerely.
What difference does direction make? If I walk in a straight line for 5 miles, then walk another 5 miles in a 50 foot circle, my starting and stopping points are only 5 miles apart, but hasn’t my person still traveled10 miles through space?
Vectors. Your motion is the sum of all the forces acting upon you. For example, in the old Doom video game your character could go slightly faster than his top forward speed by running and using the sideways Strafe movement at the same time to move at a slight diagonal.
If you’re on a train moving north at 5 mph and you are walking south at 4 mph, you are moving north at 1 mph. You are not moving in any direction at 9 mph. Velocities are vectors and add as vectors, not as scalars. If you add four vectors whose magnitudes sum to 390 mi/sec, the result is a vector with magnitude somewhere between 0 and 390 mi/sec, but you can’t know exactly without knowing the directions of all four vectors.
Last year we had a thread asking how far the Earth had traveled. Same basic problem of accounting for the different directions from the contributing components. I posted this trying to get at least a back of the envelope idea of how the motions stack up. (Trying to work out the distance traveled is harder as there are more large scale motions that contribute to aggregate distance. We can ignore them once we root the galaxy’s velocity to the CMB.)
The direction the Milky Way travels is not too far off edge on to the plane of the spiral, so where we are in the orbit around the galaxy makes a big difference to velocity. Our orbital plane around the sun is near orthogonal to the galaxy movement, so it’s contribution is more constant.
Choice of the CMB as a reference frame is probably as reasonable as one can be. So long as we are clear it is just a choice.
Thank you (all), now I understand it.
Cheers.
…then it must be an Amtrak.
Said another way, your table of various motions was about speeds, but your 5 miles straight, 5 miles in a circle example was about distances.
Speed is not the same idea as distance.
Exactly. I’d been conflating speed and distance, and slapped my forehead last night when I finally understood my error. More than a little embarrassed actually.
We’ve all done it.
This is a physics thread like every physics thread. Only two people know the answer and they have fundamental disagreements about how to express it. (Joke.)
[Letter to myself about my own experiences]
Explaining mathematical concepts in simple English words is beyond difficult at the best of times. Online threads are not the best of times. As a regular reader of popular science books about physics, I’ve found that the books that get through to me are the ones who start way back and create a foundation to build on. I’ve been presented with that foundation many times but since I don’t need it in real life for years before I read the next book, the details of the foundation easily slip away.
That’s even more true for the average reader. Physics is exceptionally difficult to translate, but the lack of a good foundation is a problem in every discipline, from English to history to economics. Every year that passes since the last encounter, which may have been a class slept through in high school, diminishes the foundation of understanding that all advanced levels build upon.
That’s one reason why simple analogies, like the bowling ball, are used and are useful. Sometimes the mechanics are far less important than grasping a concept. Why do things attract? Because they’re essentially moving downhill unless you push them up. Oh, OK. People can grasp that concept. Many of the physics threads here are actually concept threads and not mechanics threads. Once a concept is grasped, continuing on to the details of spacetime or relativity or superposition or black holes or whatever is easier.
Threads like these are great and I have equally great admiration for the experts who post here. Experts have an odd and non-obvious hurdle: they no longer can fully understand how nonexperts view their subject. Teachers face incomprehension every day and good ones learn where the blockages arise. Those who don’t encounter lack of- or mis- understanding regularly can trip over their own eagerness to get the “right” answer out.
[/Letter to myself about my own experiences]
Another, related, obstacle for teachers is that we’re usually giving a single explanation to a large audience. For any given topic in science or math (and probably in most other disciplines, as well), there are at least a half-dozen different ways to explain it. For any given student, some of those explanations will work better than others, while the wrong explanation might bore or turn off a student. And which explanation is the good one and which is the bad one will vary from student to student.
Of course, on a message board, this is compounded by the fact that you can’t see the body language of your students, and some of them might not even have been present when you started explaining.