GR: why "background-independent" and not only "background-interacting"?

Question

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?

Answer 1

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.

Answer 2

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".

Answer 3

I want to begin with this:

If spacetime were produced by matter like the electromagnetic field is produced by its sources I would understand the term "background independence".

This is an incorrect statement. The electromagnetic field exists independently of the matter sources and it is not produced by them. This is wonderfully discuted in Chap. 1 of the book Advanced Classical Electromagnetism, by Robert Wald. In electrodynamics, the sources are not sufficient to describe the full state of the electromagnetic field, and you need to also specify initial conditions. This is due to the existence, for example, of electromagnetic waves. In any electromagnetic problem, the same electromagnetic field with the addition of gravitational waves is also a solution to the Maxwell equations.

Of course, the situation in GR is similar: spacetime is not produced by its sources, it exists independently of them, although, as mentioned in other answers, it cannot be probed independently of them. Nevertheless, spacetime has degrees of freedom that are completely independent of the presence of any sources. This is greatly illustrated by the fact that if the universe is empty, gravity can still be there! Spacetime does not need to be Minkowski. Black holes and gravitational waves are vacuum solutions of general relativity: they can exist even in the absence of matter sources.

The point of background independence of GR is that it predicts the background. It does not need a pre-existing theoretical background on which it must be built. It describes the behavior of the background itself.

This is extremely different from the behavior of electromagnetism. Electromagnetism describes the evolution of the electromagnetic field and its interaction (not creation) with charged sources on top of a background that must be prescribed, in the sense that electromagnetism itself does not describe which background should be used or how the background behaves. In this sense, electromagnetism depends on the background: the background is imposed on electromagnetism and electromagnetism has no say whatsoever about it.

General relativity doesn't accept a background blindly. Rather, it describes precisely how the background spacetime must behave and evolve.

GR does not explain why spacetime exists: this is an assumption. It is background independent in the sense that the background is not imposed on GR, it is described by it.