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Landau, in his book Classical Theory of Fields, exploits the postulate that the velocity of light in a vacuum is the same in all inertial frames, to establish that the spacetime interval between two events $x^\mu=(ct,{\bf r})$ and $x^\mu+dx^\mu=(ct+cdt,{\bf r}+d{\bf r})$ defined as $$ds^2=\eta_{\mu\nu}dx^\mu dx^\nu=c^2dt^2-d{\bf r}^2\tag{1}$$ remains invariant in special relativity. In deducing this, Landau does not even mention Lorentz transformation.

Is there a similar physical postulate which enforces the invariance of $$ds^2=g_{\mu\nu}(x)dx^\mu dx^\nu \tag{2}$$ in General Relativity?

Please note that my question is not quite addressed in the answers here. JohnRennie's answer says that invariance of the metric cannot be deduced (if I understand him correctly) but then Landau derives it. Now the problem is, Landau doesn't do it in the case of GR.

So in short, I am asking is there a physical principle/postulate of general relativity that enforces (and perhaps, allows me to deduce) the requirement of the invariance of $ds^2$ in GR.

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It is inherent in the postulate that spacetime is a pseudo-Riemannian manifold with signature (-+++). That postulate brings in the whole machinery of pseudo-Riemannian geometry, including the metric and the invariance of any scalar under any arbitrary coordinate transform.

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  • $\begingroup$ Postulating spacetime to be a pseudo-Riemannian manifold implies that the metric is of the form $ds^2=g_{\mu\nu} dx^\mu dx^\nu$. But how does it mean that the metric has to be invariant under general coordinate transformation? @Dale $\endgroup$
    – SRS
    Dec 22 '20 at 18:53
  • $\begingroup$ @SRS all scalars are invariant under any coordinate transformations. That is the defining feature of a scalar. The quantity $g_{\mu\nu}dx^{\mu}dx^{\nu}$ is manifestly a scalar $\endgroup$
    – Dale
    Dec 22 '20 at 18:59
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    $\begingroup$ @SRS as I commented on your previous question, the metric is manifestly invariant under coordinate transformations. All scalar objects are. $\endgroup$
    – Eletie
    Dec 22 '20 at 19:01
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GR postulates that $ds^2$ must be locally Minkowski. In another word, in a 4D manifold, the tangent space at any point must be Minkowski. On top of that, $ds^2$ is a Lorentz scalar which makes it invariant under coordinate transformation.

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Coordinate invariance of $ds^2$ is neither a physical postulate of GR or SR. It is simply the statement that we should be able to express a physical theory in whichever coordinate system we want to express. You can see that the invariance of $ds^2$ is by design (as in it is not something non-trivial): we change $g_{\mu\nu}$ by precisely such transformation factors that they cancel out the coordinate transformation factors induced by $dx^\mu dx^\nu$.

However, there are physical postulates related to our discussion here, one for SR and one for GR.

  • The postulate of physical substance in SR is that when you make the coordinate transformation from one inertial frame to another, the metric components $\eta_{\mu\nu}$ remain invariant. Notice that this is different from just saying that $ds^2$ remains invariant. In mathematical terms, the postulate of the invariance of the metric in SR is given by $\Lambda^T\eta\Lambda=\eta$ where $\eta$ is the Minkowskian metric and $\Lambda$ is a Lorentz transformation.
  • One of the postulates of physical substance in GR is (as others have mentioned) that the metric is such that one can always find a coordinate transformation such that the metric becomes locally Minkowskian (i.e., at a point, its components are Minkowskian and its first derivatives vanish). This is related to the equivalence principle of GR.
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  • $\begingroup$ The statement that one 'should be able to express a physical theory in whichever coordinate system' is actually more along the lines of the principle of covariance. The fact that $ds^2$ is a coordinate scalar is just differential geometry. These concepts are similar but I think it important not to confuse them, especially in the context of GR. $\endgroup$
    – Eletie
    Dec 22 '20 at 19:10
  • $\begingroup$ @Eletie That is not the principle of covariance. You can write any theory in whatever coordinates you would like to write. It has nothing to do with GR. $\endgroup$
    – Dvij D.C.
    Dec 22 '20 at 19:25
  • $\begingroup$ @DvijD.C. Yes it is. It is explicitly the statement that the laws of physics should not depend on coordinates. The fact that this statement is almost vacuous in modern physics is irrelevant, and Einstein used the term bundled in with 'background independence'. It's also incorrect that it has nothing to do with GR; it can be applied to all other good theories too, but it's explicitly linked with the diffeomorphism invariance of GR, along with fact there's 'no prior geometry'. There are many many papers on this topic saying exactly this. e.g. see Norton's famous articles on the topic. $\endgroup$
    – Eletie
    Dec 22 '20 at 19:32
  • $\begingroup$ @Eletie Historically, just because GR was the first theory (or the first famous theory, I don't know) to be formulated in a coordinate independent way doesn't mean anything to the physics of the matter. It is an interesting piece of history, not physics. $\endgroup$
    – Dvij D.C.
    Dec 22 '20 at 19:42
  • $\begingroup$ @DvijD.C. considering it's still disputed as to what are the fundamental tenants of GR, and people often confuse background independence, general covariance and diffeomorphism invariance, I think it's important to be precise with these terms. It definitely is physics which theories contain background-fixed structures, or diffeomorphism breaking objects, etc. These principles are still important for looking towards theories of quantum gravity. $\endgroup$
    – Eletie
    Dec 22 '20 at 20:03
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Well, it depends on what kind of invariance that OP is talking about. Note for instance the following:

  • Any tensor, not just the metric tensor $$\mathbb{g}~=~g_{\mu\nu}~\mathrm{d}x^{\nu}\odot \mathrm{d}x^{\nu}$$ in GR, is invariant under any passive coordinate transformation.

  • A specific metric tensor $\mathbb{g}$ in GR is only invariant ${\cal L}_X\mathbb{g}=0$ under a vector field $X$ precisely if $X$ is a Killing vector field.

  • In SR the underlying manifold $M$ (i.e. Minkowski spacetime) is an affine manifold, and it make sense to ask which active affine transformations preserve the metric tensor. The latter transformations are called Lorentz transformations.

For more information, see e.g. my related Phys.SE answer here.

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"Invariance" can be an ambiguous term. If we speak of coordinate invariance, then today this invariance is understood to be somewhat trivial, and not a physical requirement at all.

The first relativists, including Einstein, thought that imposing coordinate invariance was a physical requirement. With time some doubts appeared about this; I think one of the first to point out that it wasn't a physical requirement was Fock:

See especially the Introduction (in the Preface he writes that among his purposes is "to correct a widespread misinterpretation of the Einsteinian Gravitation Theory as some kind of general relativity").

With the subsequent developments of general relativity and also of differential geometry it became understood that coordinate invariance has no physical content. It is in fact possible in principle to present differential geometry and relativity in a coordinate-free formalism; for an example see Misner, Thorne, Wheeler below and also

who uses the coordinate and coordinate-free formalisms side by side.

This fact became even clearer when it was shown that Newtonian mechanics could also be presented in a 4-dimensional, coordinate-invariant way. When presented this way, Newtonian gravitation is called Newton-Cartan theory, since it was formalized this way mainly by Cartan.

Coordinate-invariant spacetime presentations of Newtonian mechanics can for example be found in:

...and many others.

I'd recommend reading these works and seeing how Newtonian mechanics can be expressed similarly to general relativity, because this helps understanding the real physical differences between the two theories.

So if we mean "coordinate invariance" the answer to your question is that no physical postulate requires it: it's a feature of the differential-geometric formalism.


What then is the physical difference between Newtonian mechanics, special relativity, and general relativity? Here is where the word "invariance" is sometimes used with a different meaning.

All three theories are represented by a metric field (with different signature depending on the theory) on a 4D spacetime manifold. In Newtonian mechanics and special relativity this field is invariant with respect to specific groups of transformations of the manifold (different groups for the two theories). Figuratively speaking, we can imagine to "move" the manifold and overlap it with itself in a different position. The metric field will then perfectly overlap with itself. In other words, the geometry ofspacetime itself has specific symmetries in these two theories. This is not coordinate invariance, although it leads to the existence of particularly simple coordinate charts. In general relativity no such symmetries exist, in general, for spacetime, because it lets the metric be a dynamic field. But some specific solutions of the theory can have symmetries (this connects to Killing vectors).

So if we speak of this kind of spacetime invariance, the answer to your question is that general relativity actually requires no such invariance a priori.

The metric field is not the only relevant physical/geometric object, however: a similar discussion can be made about the affine connection, which both in Newtonian gravitation and general relativity expresses the gravito-inertial field.

The term "covariance" is also sometimes used to try to disentangle these two meanings (and then it has a different meaning from "covariance" as opposed to "contravariance"). It seems to me that the use of these terms is quite varied in the literature, so it's always best to make sure what an author means by them.

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