What's the underlying frame of the Einstein's Field Equation?

[Pyter]

By "observed mass" I meant "relativistic mass", but since I've just learned that it's no longer a thing, I retract.

 

[Pyter]

Coordinates are arbitrary "maps" of spacetime points, which do not have a priori any physical meaning, because they change under general (local) coordinate transformations

About that, I've found online a very simple and satisfying lecture directly from the horse's mouth: Gaussian coordinates explained by Einstein.
In this chapter, he also explains why they're enough to describe any physical event.

 

[Orodruin]

About that, I've found online a very simple and satisfying lecture directly from the horse's mouth: Gaussian coordinates explained by Einstein.
In this chapter, he also explains why they're enough to describe any physical event.

General pro tip: Modern textbooks will usually do a better job in terms of pedagogy, didactics, and up to date treatment than anything written by Einstein.

 

[Pyter]

General pro tip: Modern textbooks will usually do a better job in terms of pedagogy, didactics, and up to date treatment than anything written by Einstein.

I've noticed, the chapter before that where he makes the analogy with the marble table and the metal rods is quite confusing, I couldn't tell if the local temperature changes only affected the rods or also the table.
But the chapters I've cited helped me "visualize" the physical meaning of the GR, because that particular book is written in layman terms, without too much math and topology.

 

[vanhees71]

When it comes to relativistic thermodynamics and statistical physics, anything written before 1968 is confusing, because the foundations haven't been understood before van Kampen et al. Nowadays, every intrinsic property (among them the intensive quantities of thermodynamics like, temperature, chemical potential, densities of various other thermodynamical potential, and entropy density) related to some medium are defined as scalar (field) quantities as measured in the (local) rest frame of a "fluid cell".

 

[Pyter]

What would be measured by a radar measurement’s round trip time (multiplied by c/2) would be
$$
\int \sqrt{-\frac{g_{rr}}{g_{tt}}} dr = \int g_{rr} dr.
$$

Did you get that formula from the metric $ds^2 = g_{tt}\, (c\,dt)^2 - g_{rr}\,dr^2$, by setting $ds^2 = 0$ for the light-like path?

 

[Orodruin]

Did you get that formula from the metric $ds^2 = g_{tt}\, (c\,dt)^2 - g_{rr}\,dr^2$, by setting $ds^2 = 0$ for the light-like path?

Yes. What you want is to integrate dt to obtain the global time difference.

Note: It is $ds^2 = g_{tt} dt^2 + g_{rr} dr^2$. The signs are included in the metric components.

 

[Pyter]

Note: It is $ds^2 = g_{tt} dt^2 + g_{rr} dr^2$. The signs are included in the metric components.

Right. But not $c^2$, I guess? Otherwise either the RHS of the line element is not dimensionally correct, or $g_{tt}$ has a dimension different from $g_{rr}$.

 

[Orodruin]

Right. But not $c^2$, I guess? Otherwise either the RHS of the line element is not dimensionally correct, or $g_{tt}$ has a dimension different from $g_{rr}$.

I always use units with $c = 1$. It is just more ... natural.

 

[cianfa72]

Yes. What you want is to integrate dt to obtain the global time difference.

Note: It is $ds^2 = g_{tt} dt^2 + g_{rr} dr^2$. The signs are included in the metric components.

So for the round-trip coordinate time $\Delta t$ of the complete null path $ds=0$ from the spaceship hovering at fixed $(\theta, \phi, r_1)$ to the star surface at $(\theta, \phi, r_0)$ and back we get the value of that integral multiplied by 2.

 

[Orodruin]

Yes

 

[Pyter]

I always use units with $c = 1$. It is just more ... natural.

Of course, I was talking about MKS units.