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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?

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    $\begingroup$ 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. $\endgroup$
    – Prahar
    Nov 9, 2023 at 21:56
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    $\begingroup$ Using your analogy, the background independence refers to the fact that the artist can create “any possible object” using clay. $\endgroup$
    – Prahar
    Nov 9, 2023 at 21:58
  • $\begingroup$ It is also synonymous with the phrase "no prior geometry", which is perhaps more obvious as to its meaning. $\endgroup$
    – Eletie
    Nov 10, 2023 at 9:13
  • $\begingroup$ @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... $\endgroup$
    – MartyMcFly
    Nov 10, 2023 at 10:04
  • $\begingroup$ @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). $\endgroup$
    – Eletie
    Nov 10, 2023 at 10:26

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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.

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  • $\begingroup$ 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. $\endgroup$
    – MartyMcFly
    Nov 10, 2023 at 19:30
  • $\begingroup$ @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. $\endgroup$
    – JanG
    Nov 10, 2023 at 19:42
  • $\begingroup$ However, Minkowski spacetime as a mathematical object influences physics: It's the limit in infinity if only one central mass is calculated. $\endgroup$
    – MartyMcFly
    Nov 10, 2023 at 19:50
  • $\begingroup$ No, asymptotically flat spacetime means only that the Riemann curvature tensor will be arbitrarily small in limit but never zero. $\endgroup$
    – JanG
    Nov 10, 2023 at 19:55
  • $\begingroup$ 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. $\endgroup$
    – MartyMcFly
    Nov 10, 2023 at 20:24
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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".

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