# I Issue With Derivation of Gravitational Time Dilation

#### bsalha

Summary
Why does the derivation of gravitational time dilation use classical mechanics' definition of kinetic energy as opposed to the relativistic definition of kinetic energy?
Why do we use the equation $\frac {1}{2}mv^2 = \frac {GmM}{r}$ to derive potential velocity, and then put that in the Lorentz factor in order to derive gravitational time dilation? Shouldn't we be using the relativistic definition of kinetic energy -> $mc^2(\gamma - 1)$ to derive the gravitational time dilation? Using the first approach we get $$\gamma = \frac 1 {\sqrt {1 - \frac {2GM} {rc^2}}},$$ but using the relativistic definition of kinetic energy we get $$\gamma = \frac {GM} {rc^2}+1$$, so clearly one of these approaches is wrong.

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#### PeterDonis

Mentor
Why do we use the equation $\frac {1}{2}mv^2 = \frac {GmM}{r}$ to derive potential velocity, and then put that in the Lorentz factor in order to derive gravitational time dilation?
Where have you seen gravitational time dilation derived this way? Can you give a reference?

#### Ibix

Why do we use the equation $\frac {1}{2}mv^2 = \frac {GmM}{r}$ to derive potential velocity, and then put that in the Lorentz factor in order to derive gravitational time dilation?
As Peter is hinting, your derivation is incorrect. $v$ in your calculation is the Newtonian velocity of something dropped from rest at infinity to radius $r$. But the gravitational time dilation formula you quote applies between hovering observers - who would regard such infalling objects as moving, so a different time dilation factor would apply. Even if the Newtonian calculation of the velocity were correct, which it's not.

Furthermore, $\gamma$ is the kinematic time dilation factor between inertial frames, so can only be used locally in general relativity. It makes no sense to apply it to comparing rates of distant clocks in curved spacetime because $v$ cannot be defined the way it requires.

Peter asked for a reference. If you can provide one, we'd be interested because it's a source to avoid. If, on the other hand, it's something you came up with yourself, you've simply found an amusing coincidence, not a derivation.

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#### Dale

Mentor
Why does the derivation of gravitational time dilation use classical mechanics' definition of kinetic energy as opposed to the relativistic definition of kinetic energy?
That is interesting. I know of two different derivations, and neither uses KE at all. The ones I am familiar with either use the equivalence principle or directly from the metric.

If you don’t like that derivation then I would just find another one that you do like

#### bsalha

Where have you seen gravitational time dilation derived this way? Can you give a reference?
I did not find it online, someone told me about it.

#### bsalha

I knew this derivation made no sense because the energy principle in this scenario is applying to an energy conserved system consisting of an infinite amount of space between the observer and the object and it makes so many assumptions that I can't even comprehend. However it is quite insane that the approximation for kinetic energy in the system can be used in for the velocity in the Lorentz factor and one can get the expression for gravitational time dilation.

#### PeterDonis

Mentor
Did this someone give you an actual reference? Or were they just making it up?

As you can see from the responses you have gotten in this thread, there is no such derivation.

it is quite insane that the approximation for kinetic energy in the system can be used in for the velocity in the Lorentz factor and one can get the expression for gravitational time dilation
No, it's not "insane", it's just a coincidence, as @Ibix has pointed out. What's more, the coincidence is even less impressive than it might seem at first glance. Consider: the kinetic energy expression includes a velocity $v$. But what object has this velocity? Certainly not the object whose gravitational time dilation is equal to the expression you give, because that object is static--that expression for gravitational time dilation is for an object which is at rest relative to the gravitating mass, at radial coordinate $r$ (which is not quite the same as the actual proper distance of the object from the center of the gravitating mass). So the coincidence is the result of mixing together two formulas that don't even apply to the same object.

#### bsalha

I know it is a coincidence I'm just saying that it is funny that you can incorrectly come to a correct conclusion by failing to be precise. And no this person didn't have a source he just attempted to oversimplify the derivation then I asked why he used the approximation for kinetic energy and he said you just assume you're not going that fast and then I got even more confused.

#### PeterDonis

Mentor
this person didn't have a source he just attempted to oversimplify the derivation
He wasn't oversimplifying a valid derivation to begin with: the entire reasoning is invalid because, as I said, the $v$ in the kinetic energy formula can't possibly apply to the object whose gravitational time dilation is given by the formula that is derived. The same would be true if you tried to use the relativistic formula for kinetic energy.

#### Ibix

I know it is a coincidence I'm just saying that it is funny that you can incorrectly come to a correct conclusion by failing to be precise.
This is essentially numerology, just with algebra instead of arithmetic. Like numerology, coincidences come up more often than you might naively think. Like numerology, there's no meaning to the coincidences.

"Issue With Derivation of Gravitational Time Dilation"

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