About the Equivalence Principle

[vanhees71]

In the non-relativistic limit the pseudo-metric reads
$$g_{00}=1-\frac{2MG}{r}, \quad g_{0j}=g_{j0}=0, \quad g_{ij}=-\delta_{ij}.$$
Here, latin indices run from 1 to 3, and I set $c=1$. From this you get
$$\mathrm{d} \tau^2 = \left (1-\frac{2MG}{r} \right) \mathrm{d} t^2-\mathrm{d} \vec{x}^2, \quad r=|\vec{x}|$$
A clock's proper time is thus
$$\mathrm{d} \tau = \mathrm{d} t \sqrt{1-\frac{2MG}{r}} \simeq \mathrm{d} t \left (1-\frac{MG}{r} \right).$$
The coordinate time $t$ is the time a clock very far from the mass $M$ (which is located around the origin of the spatial coordinate system).

 

[JohnnyGui]

In the non-relativistic limit the pseudo-metric reads
$$g_{00}=1-\frac{2MG}{r}, \quad g_{0j}=g_{j0}=0, \quad g_{ij}=-\delta_{ij}.$$
Here, latin indices run from 1 to 3, and I set $c=1$. From this you get
$$\mathrm{d} \tau^2 = \left (1-\frac{2MG}{r} \right) \mathrm{d} t^2-\mathrm{d} \vec{x}^2, \quad r=|\vec{x}|$$
A clock's proper time is thus
$$\mathrm{d} \tau = \mathrm{d} t \sqrt{1-\frac{2MG}{r}} \simeq \mathrm{d} t \left (1-\frac{MG}{r} \right).$$
The coordinate time $t$ is the time a clock very far from the mass $M$ (which is located around the origin of the spatial coordinate system).

Thanks. I've still yet to delve into pseudo-metrics and its formula derivations.

I've got 2 questions regarding your formula:

1. Is the Equivalence Principle explanation some kind of alternative substitute for the gravitational potential explanation in case of a local scenario?
2. If so, does this mean that if you apply the gravitational potential formula in your post for a local scenario, it would dilate the time by the same factor the Equivalence Principle would? Thus, time dilated by a infinitesimal small change in g (local) would be approximately equivalent to the time dilated by the difference in velocities?

 

[jbriggs444]

2. If so, does this mean that if you apply the gravitational potential formula in your post for a local scenario, it would dilate the time by the same factor the Equivalence Principle would? Thus, time dilated by a infinitesimal small change in g (local) would be approximately equivalent to the time dilated by the difference in velocities?

It's not the change in g over distance. It's the integral of g over distance.

 

[JohnnyGui]

It's not the change in g over distance. It's the integral of g over distance.

Sorry, that's what I meant in the question.

 

[PeterDonis]

I can understand time dilation if it's explained mathematically by the difference in velocity

Which only works locally, in a small enough patch of spacetime such that there is no difference in g within the patch. So this explanation doesn't help you over a large enough range of height that differences in g are observable.

how can time dilation be explained by the difference in gravity potential?

Because that's how the math works. It's different math from the math you're using in a local inertial frame, but it still works.

To put it another way, the time dilation between observers at different heights in a gravity well is simply part of the geometry of spacetime in this particular scenario. That is the explanation.

Especially when the equivalence principle can't be used for time dilation over a large range of heights?

The equivalence principle can't tell you anything about the geometry of spacetime anyway, because by definition it only works in a small enough patch of spacetime that tidal gravity (which is the physical observation referred to by "the geometry of spacetime") is unobservable.

Basically, you're looking at things backwards. The geometry of spacetime is the fundamental entity in GR; the equivalence principle is just one of many possible approximations. You're trying to make the EP the fundamental entity and derive everything else from it. That won't work.

 

[PeterDonis]

Is the Equivalence Principle explanation some kind of alternative substitute for the gravitational potential explanation in case of a local scenario?

No. It's just a different way of looking at the same explanation (or a local approximation of it). See my post #29.

does this mean that if you apply the gravitational potential formula in your post for a local scenario, it would dilate the time by the same factor the Equivalence Principle would?

The gravitational potential formula is exact and applies over any height change, however small. The local EP formula is an approximation to it that only works for small enough height changes that no change in g is observable. So you are once again looking at it backwards: the question isn't whether the potential formula gives the same answer as the EP formula, the question is whether, and under what condtions, the EP formula (the approximation) gives the same answer (to within the desired accuracy) as the potential formula (the exact formula).

 

[JohnnyGui]

The equivalence principle can't tell you anything about the geometry of spacetime anyway, because by definition it only works in a small enough patch of spacetime that tidal gravity (which is the physical observation referred to by "the geometry of spacetime") is unobservable.

Basically, you're looking at things backwards. The geometry of spacetime is the fundamental entity in GR; the equivalence principle is just one of many possible approximations. You're trying to make the EP the fundamental entity and derive everything else from it. That won't work.

Yes, what you describe here was how I was considering EP previously. But I have already noticed and am aware that it only works in a small enough patch of spacetime which made me ask how gravitational potential mathematically proves time dilation. Vanhees71 replied to that but I still have yet to understand this different math problem.

The gravitational potential formula is exact and applies over any height change, however small. The local EP formula is an approximation to it that only works for small enough height changes that no change in g is observable. So you are once again looking at it backwards: the question isn't whether the potential formula gives the same answer as the EP formula, the question is whether, and under what condtions, the EP formula (the approximation) gives the same answer (to within the desired accuracy) as the potential formula (the exact formula).

Ah, ok. So in case of an infinitesimal small g change, EP would give a sufficient approximation for the potential formula?

I think an analogue for what EP is for the Potential formula is what Newton's formulas are for relativistic ones.

 

[PeterDonis]

in case of an infinitesimal small g change, EP would give a sufficient approximation for the potential formula?

Basically, yes.

I think an analogue for what EP is for the Potential formula is what Newton's formulas are for relativistic ones.

Not really. The Newtonian formulas are approximations to the relativistic ones that hold in a particular kind of spacetime, in which there is an isolated source of gravity surrounded by empty space, and in which the source of gravity is weak and all objects are moving slowly compared to the speed of light.

The EP is valid in a sufficiently small patch of any spacetime whatsoever; there is no limitation that gravity be weak or that objects be moving slowly. The only limitation is that the patch of spacetime be small enough that tidal gravity is not observable.

 

[vanhees71]

The equivalence principle is a heuristic principle that motivates the formulation of GR in terms of spacetime geometry. Mathematically it boils down to the assumption that spacetime is a pseudo-Riemannian manifold with a pseudometric of signature (1,3) (or equivalently (3,1)) and that test-point particles follow time-like (or null for "photons") geodesics if there's no other force than gravity.