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- TL;DR Summary
- Explaining Dirac's assertion ("GTR", Ch. 16) that in a static gravitational field we must have ##g_{m0} = 0, m=1,2,3)##.

In Dirac's "General Theory of Relativity", he begins Chap 16, with

It's obvious that static ##\rightarrow g_{\mu\nu,0}=0##, but

What I can think of is this. First, think of ordinary 3D space. Suppose there is no curvature in one of the dimensions, say the ##x^1## dimension. Then the metric ##ds^2 = g_{mn} dx^m dx^n## should have no ##dx^1 dx^2## or ##dx^1 dx^3## terms, since translating along the ##x^1## coordinate direction should not alter how ##ds^2## depends upon ##x^2## or ##x^3##.

In the same way, there should be no ##dx^0 dx^m## terms in the metric if the curvature of spacetime is static in time.

Alternatively, if I make the change of coordinates ##x'^0=x^0+\text{constant}##, and ##x'^m=x^m## (##m=1,2,3##), and if the gravitational field is

Is this the right way to explain Dirac's assertion?

*"Let us consider a static gravitational field and refer it to a static coordinate system. The ##g_{\mu\nu}## are then constant in time, ##g_{\mu\nu,0}=0##. Further, we must have ##g_{m0} = 0, (m=1,2,3)##."*It's obvious that static ##\rightarrow g_{\mu\nu,0}=0##, but

**why must ##g_{m0} = 0## ?**What I can think of is this. First, think of ordinary 3D space. Suppose there is no curvature in one of the dimensions, say the ##x^1## dimension. Then the metric ##ds^2 = g_{mn} dx^m dx^n## should have no ##dx^1 dx^2## or ##dx^1 dx^3## terms, since translating along the ##x^1## coordinate direction should not alter how ##ds^2## depends upon ##x^2## or ##x^3##.

In the same way, there should be no ##dx^0 dx^m## terms in the metric if the curvature of spacetime is static in time.

Alternatively, if I make the change of coordinates ##x'^0=x^0+\text{constant}##, and ##x'^m=x^m## (##m=1,2,3##), and if the gravitational field is

**, then ##g'_{\mu\nu}=g_{\mu\nu}## (because the time translation cannot alter spacetime intervals). Hence, ##g_{\mu\nu} dx'^\mu dx'^\nu = g_{\mu\nu} dx^\mu dx^\nu##, which implies there are no ##dx^0 dx^m## terms in the metric.***static*Is this the right way to explain Dirac's assertion?

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