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I know how to derive the lorentz time dilation equation. I am wondering how to derive the equation for gravitational time dilation: T=T_{o}(1/(sqrt(1-(2GM)/(Rc^{2})))
The way you have it written, the formula applies only to the Schwarzschild solution, where R is the Schwarzschild radius. In that case, the way to derive it is to write down the Schwarzschild metric, and look at g_{00}. For a motionless test particle (dr = dθ = dφ = 0),I know how to derive the lorentz time dilation equation. I am wondering how to derive the equation for gravitational time dilation: T=T_{o}(1/(sqrt(1-(2GM)/(Rc^{2})))
You do if the expression involves the Schwarzschild radius.I have a treatment in my SR book, http://www.lightandmatter.com/sr/ , in ch. 5. You don't need GR or the Schwarzschild metric.
It can be expressed in terms of the gravitational potential.You do if the expression involves the Schwarzschild radius.
I don't understand what you mean by "##R## is the Schwarzchild radius". I would expect that ##R## is just the radial coordinate, in his notation. But I agree with the 'derivation'. Just assume Schwarzschild metric, and then choose a stationary test object outside the event horizon, then you get the time dilation formula. And this is often used as an approximation for the general relativistic equation for the time dilation due to planets, I think. So it's a good metric to use in most of the simple textbook problems on GR time dilation.The way you have it written, the formula applies only to the Schwarzschild solution, where R is the Schwarzschild radius. In that case, the way to derive it is to write down the Schwarzschild metric, and look at g_{00}. For a motionless test particle (dr = dθ = dφ = 0),
ds^{2} = (1 - 2GM/Rc^{2}) dt^{2},
and from this, ds/dt gives you the time dilation.
pretty cool. I always assumed that anyone that posts on physicsforums wouldn't have time to write a book along with doing their full-time job. hehe. you must have good time-management skills. In the discussion of gravitational potential in chapter 5, I'm not totally sure what you mean though. I understandbcrowell said:I have a treatment in my SR book, http://www.lightandmatter.com/sr/ , in ch. 5. You don't need GR or the Schwarzschild metric.
It's true in general, not just in the Newtonian limit, that [itex]df/f=-d\Phi[/itex]. If you integrate both sides, you get [itex]f\propto e^{-\Phi}[/itex], which holds for any static spacetime, including the Schwarzschild spacetime.Anyway, that's kinda interesting. Although I suppose it only works in the Newtonian limit.
If R is some other radial coordinate and not the Schwarzschild radial coordinate, then the formula is false.I don't understand what you mean by "##R## is the Schwarzchild radius". I would expect that ##R## is just the radial coordinate, in his notation.
Are you sure about that? I'm reading the formula as applying to any radius larger than the Schwarzschild radius. At the Schwarzschild radius the formula produces infinite red-shift.If R is some other radial coordinate and not the Schwarzschild radial coordinate, then the formula is false.
cool. Is ##\Phi## one of the Hansen potentials? (the mass potential). I just read this on the wikipedia page for stationary spacetime. That's some pretty interesting stuff. So.. it seems like the Newtonian gravitational potential has a (perfect?) analogue in stationary spacetimes in GR. It's not often that a concept carries on so nicely from Newtonian gravity to GR.It's true in general, not just in the Newtonian limit, that [itex]df/f=-d\Phi[/itex]. If you integrate both sides, you get [itex]f\propto e^{-\Phi}[/itex], which holds for any static spacetime, including the Schwarzschild spacetime.
As long as you're in a space-time with a time-like Killing field ##\xi^{\mu}## you can always define a gravitational potential by ##\Phi = \frac{1}{2}\ln (-\xi_{\mu}\xi^{\mu})##. It's clear why this makes sense physically as observers following orbits of ##\xi^{\mu}## are at rest in the gravitational field, or equivalently the asymptotic Lorentz frame if the space-time is asymptotically flat, and their 4-acceleration is simply ##a^{\mu} = \nabla^{\mu}\Phi## which is (the negative of) the gravitational acceleration of freely falling particles with respect to the stationary ("Copernican") frames defined by ##\xi^{\mu}##. If ##\xi_{[\gamma}\nabla_{\mu}\xi_{\nu]} \neq 0## then we can also write down Newtonian analogues of the centrifugal and Coriolis accelerations of freely falling particles relative to the stationary frames.So.. it seems like the Newtonian gravitational potential has a (perfect?) analogue in stationary spacetimes in GR.