General relativity and time dilation

[Joe1]

Hello,
I have a question about general relativity and time dilation. I found an expression for the time dilation around a sphere of radius R and mass m.
T=T0/(sqrt(1-2Gm/Rc^2))
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html
Is there a way to calculate the time dilation for a different structure directly from the potential? Can I just substitute my potential in for Gm/R in the equation above?

Thanks,
Joe

 

[pervect]

Basically, yes, as long as you are in a "weak field" approximation. "Weak field" would include anything in the solar system.

Note that in the weak field, T0/sqrt(1-2GM/rc^2) is essentially equal to
T0*(1+GM/rc^2)

 

[pmb_phy]

Hello,
I have a question about general relativity and time dilation. I found an expression for the time dilation around a sphere of radius R and mass m.
T=T0/(sqrt(1-2Gm/Rc^2))
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html
Is there a way to calculate the time dilation for a different structure directly from the potential? Can I just substitute my potential in for Gm/R in the equation above?

Thanks,
Joe

In general relativity the term "potential" has a different expanded usage. The 10 independent components of the metric tensor are the potentials of the gravitational field. That's why the metric is sometimes referred to as the tensor potential. The general expression for time dilation is

dtau = sqrt(g₀₀)dt

Pete

 

[pervect]

Pete's response is at cross purposes from mine.

While some people do call the metric coefficients the "gravitational potential", I find that it is less confusing to just call the metric coefficients the metric coefficients.

The potential I was talking about was the Newtonian potential.

In the weak field approximation, time dilation is just 1+U/c^2, where U is the negative of the Newtonian potential,

i.e U = GM₁/R₁+GM₂/R[2]...+GM_n/R_n

This will be the same as 1/sqrt(1-2U) in any situation in which the approximation is valid.

Here the M_i are the masses, and the R_i are the distances that a point P is away from the masses.

This is also called the PPN approximation.

The PPN approximation requires non-relativistic velocities as well as a weak field, but is routinely used for problems in the solar system.

If you have a general metric, you can always use sqrt(|g₀₀|) or it's reciprocal as the time dilation factor. (The reciprocal factor depends on whether one wants a number lower than 1 to represent a slower clock or a number larger than 1 to represent a slower clock - the first form uses the first convention, the reciprocal form uses the second convention). Stationary clocks in a gravity well always tick more slowly than stationary clocks at infinity.

The problem with this approach in general is that finding the metric for an arbitrary configuration is not an easy task. Fortunately strong field situations are rare, and the PPN approximation works fine for solar-system problems.

 

[Joe1]

ok, so T=T0/(sqrt(1-2Gm/Rc^2)) is an approximation for weak gravitational fields, but for massive objects it is necessary to use the equation that Pete mentioned. How would I find g00? Can this be determined directly from the potential?

Thanks for your help,
Joe

 

[masudr]

g00 is a component of the metric tensor. The metric tensor satisfies the gravitational field equation; one must therefore solve the field equation for the 00 component. This is, in general, quite a difficult thing to do. However, for some simple systems people have solved it (i.e. Schwarzschild solution etc.)

 

[pervect]

ok, so T=T0/(sqrt(1-2Gm/Rc^2)) is an approximation for weak gravitational fields, but for massive objects it is necessary to use the equation that Pete mentioned. How would I find g00? Can this be determined directly from the potential?

Thanks for your help,
Joe

T=T0/(sqrt(1-2Gm/Rc^2))

is actually exact, because it's in the form T = T0 / sqrt(|g00|).

Look at for instance
http://en.wikipedia.org/wiki/Schwarzschild_metric

and see that g00 for the Schwarzschild metric is just the coefficient of c^2 dt^2, i.e.

g00 = -1 + 2GM/Rc^2

Thus the formula you cite is exact for the Schwarzschild metric, which is the exact solution for a single massive body.

If you have a single massive object, this above formula is good. If you have more than one massive object, to get an exact solution you'd have to solve Einstein's field equations to find the metric. Analytic solutions are only known for 1 body, so you'd have to solve Einstein's equations numerically. This is an extremely difficult task. Therfore people use the approximation I mentioned.

Talking about "the potential" in GR is just going to confuse the issue, I'm afraid.

Talking about the Newtonian potenital makes sense in the Newtonian approximation, which is fine for practical, solar-system purposes. Talking about "the potential" in GR is making a conceptual error. GR doesn't have a potential in the same sense that Newtonian gravity does.