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My question is closely related to the answer of this question:

Why is general relativity background independent and electromagnetism is background dependent?

General Relativity is often stated to be "background independent", because it calculates how spacetime is curved. That's in contrast to other theories like classical electrodynamics which act on the manifold without interacting with it.

I understand that GR interacts with spacetime (and that that's a great advancement in comparison to the theories before) - however, I do not understand, why it's called "background independent". Matter curves spacetime. That means to me that there has to be a spacetime first which can be curved by matter.

If the universe is empty (in GR), Minkowski spacetime is still there. If spacetime were produced by matter like the electromagnetic field is produced by its sources I would understand the term "background independence". But to me it's only interacting with the background, not background independent. (Like an artist who is forming any possible object out of clay is not independent of clay... Just acting on it in every possible way)

How shall I understand the term "background independence"? Is the term not precise? Did I get something wrong?

Qmechanic
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MartyMcFly
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    The “background” being referenced is not spacetime. It’s the geometry of spacetime. GR has the same form whether or not the background spacetime is Minkowski, Schwarzschild, Kerr, etc. In that sense its background independent. – Prahar Nov 09 '23 at 21:56
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    Using your analogy, the background independence refers to the fact that the artist can create “any possible object” using clay. – Prahar Nov 09 '23 at 21:58
  • It is also synonymous with the phrase "no prior geometry", which is perhaps more obvious as to its meaning. – Eletie Nov 10 '23 at 09:13
  • @Eletie the "prior geometry" of GR is Minkowski space. Why do you say that Minkowski space is "no prior geometry"? I don't understand that. But many are arguing like this... – MartyMcFly Nov 10 '23 at 10:04
  • @MartyMcFly no prior geometry is simply the expression that in GR, the geometry is not fixed a priori (e.g., it's dynamical). Contrast this to Special Relativistic theories, where the Minkowski metric is fixed a priori (non dynamical). – Eletie Nov 10 '23 at 10:26
  • @Eletie Would you agree that in GR the geometry is fixed to be locally the SR geometry? – JanG Nov 12 '23 at 12:05
  • @JanG no, locally pseudo-Euclidean is a property of all differentiable manifolds. This is not the same having a fixed, non-dynamical metric $\eta$ as in SR – Eletie Nov 12 '23 at 17:39
  • @Eletie I can see what you mean, I agree. However, what is the reason why GR is locally (in normal coordinates) Minkowski: is it the demand of constancy of light velocity? – JanG Nov 12 '23 at 18:48
  • @JanG The speed of light being in constant only holds in SR, so that would be circular reasoning. Any theory based on differential manifolds has a flat tangent space – Eletie Nov 12 '23 at 21:15
  • @Eletie It is more about metric signature in that tangent space. I came across similar view on this forum. My argument is: in our labors (so local in earth gravitation field) in experiments with electromagnetic and gravitation waves we use the fact of absoluteness of physical constant $c$ (SR). – JanG Nov 13 '23 at 07:24
  • @JanG Well GR is constructed in order to reproduce SR within its regime of validity, so it must be the case that we have a Lorentzian signature. This doesn't at all seem related to the concept of GR having a dynamical metric though. – Eletie Nov 13 '23 at 09:47
  • @Eletie I agree with both your sentences. The do not contradict each other in my view. The requirement of Lorentzian signature comes from physical experiment. It is some kind of boundary condition for GR mathematical theory as (3+1)-dimensional pseudo-Riemannian manifold. – JanG Nov 13 '23 at 11:26

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Your question deserves more eloquent answer but maybe you will find my try useful for your understanding.

The General relativity describes gravitation as pseudo-Riemannian locally Lorenztian (Minkowski) manifold called spacetime. It is not generated by matter. Matter for gravitation is merely a distortion of its geometry. On the other hand, a warped spacetime changes the energistic properties of matter (tensor T).

An empty universe is an oxymoron. Without matter no one can state that universe is empty (you are matter). Otherwise, matter without space and time would be not able to manifest itself. Can you imagine matter without space and time? The spacetime and matter are related to each other like Yin & Yang.

Asked about difference between Newton’s and his gravitation theory Einstein allegedly said: “It was formerly believed that if all material things disappeared, time and space would be left. According to the relativity theory, however, time and space disappear together with the things.”

The more scientific reference provides mathematics. Einstein field equations are second order differential equations on metric. The trivial solution g$=0$ describes exactly what Einstein said namely that without matter there is no spacetime.

Rephrasing your 'artist and clay' comparison – in case of spacetime the clay is the artist.

JanG
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  • Thank you!! This citation of Einstein I heard before. However, as we calculate temporarily with GR, spacetime seems to exist without matter: There's flat, Minkowskian spacetime witbout matter. – MartyMcFly Nov 10 '23 at 19:30
  • @MartyMcFly What you mean is Minkowski spacetime as mathematical object. In mathematics you can always postulate some space or mathematical object. In physics we deal with physical reality which is the spacetime with matter (somewhere). Spacetime is Minkowski only local. – JanG Nov 10 '23 at 19:42
  • However, Minkowski spacetime as a mathematical object influences physics: It's the limit in infinity if only one central mass is calculated. – MartyMcFly Nov 10 '23 at 19:50
  • No, asymptotically flat spacetime means only that the Riemann curvature tensor will be arbitrarily small in limit but never zero. – JanG Nov 10 '23 at 19:55
  • No, the limit is exactly Minkowski (not "arbitrarily small cuvature tensor"). It will never be reached. - - - However, what's important is that it's just a background which exists independent of the matter. – MartyMcFly Nov 10 '23 at 20:24
  • An asymptote is per definition something that is never reached. But never mind. If you persist on that background independent of matter, it is your choice. – JanG Nov 11 '23 at 06:52
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I would say more that General relativity is "sometimes" stated to be background independent. This is because it is a somewhat clumsy formulation and most textbooks will avoid it. The more accurate statement is that it makes dynamical a structure (space-time geometry) that is otherwise taken as "guaranteed" or simply "as a background". How this works is already aptly explained by Prof. Legolasov in the original answer. In short, one does not have to assume much about the space-time geometry, it is determined completely by the theory. (It is perhaps a bit weak to say that GR only "interacts" with the background, it is the theory of the dynamics of the background.)

There are other structures that GR takes for granted in a typical configuration. Some of them are topology, differential structure, signature of the metric, or the number of dimensions for instance. Call these the "background", and suddenly you can say that GR is also "background-dependent".

Void
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