# Gravitational time dilation and special relativity time dilation

#### yakmastermax

Looking at the two equations for time dilation they seem very similar

$$t_{surface} = t_{space} \sqrt{1-\frac{2GM}{rc^2}}$$
$$t_{moving} = t_{observer}\sqrt{1-\frac{v^2}{c^2}}$$

I was hoping someone could explain more how they are connected?

I'd like to think that a fast moving object with β near 1 would begin to feel more massive and inertial. Being more inertial the object would then "slow down" as described by general relativity? Is the special relativity expression an extension of general relativity or are the two separate?

Thanks all!

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#### PeterDonis

Mentor
I was hoping someone could explain more how they are connected?
They aren't. They are expressions for two different things.

Is the special relativity expression an extension of general relativity or are the two separate?
They are separate.

#### yakmastermax

If you solve for the velocity though you get the expression for escape velocity.
$$v_e=\sqrt{\frac{2GM}{r}}$$

Forgive me for being stubborn, I'm just trying to connect dots

#### PeterDonis

Mentor
If you solve for the velocity though you get the expression for escape velocity.
$$v_e=\sqrt{\frac{2GM}{r}}$$
Yes, that's true, but it doesn't mean the two formulas are different ways of expressing the same thing.

The first formula gives the time dilation of a clock at rest at radius r, relative to a clock at rest at infinity.

The second formula gives the time dilation of a clock moving at speed v, relative to a clock at rest.

Equating the two formulas therefore doesn't really make sense: one formula refers only to clocks at rest, the other requires one clock to be moving. The fact that v = escape velocity comes out is just a coincidence.

Forgive me for being stubborn, I'm just trying to connect dots
No problem, that's what PF is here for!

#### pervect

Staff Emeritus
Looking at the two equations for time dilation they seem very similar

$$t_{surface} = t_{space} \sqrt{1-\frac{2GM}{rc^2}}$$
$$t_{moving} = t_{observer}\sqrt{1-\frac{v^2}{c^2}}$$

I was hoping someone could explain more how they are connected?

I'd like to think that a fast moving object with β near 1 would begin to feel more massive and inertial. Being more inertial the object would then "slow down" as described by general relativity? Is the special relativity expression an extension of general relativity or are the two separate?

Thanks all!
There's nothing super-obvious that stands out, except that both expressions are in the form

(time dilation ratio)^2 + (something else)^2 = 1

2GM/c^2r is always positive, so it's reasonable to replace it with a square root.

It's sort of suggestive, but as I say, nothing obvious stands out as far as any more direct relationship.

#### yakmastermax

I guess the conclusion I'm hoping to draw is that the two types of time dilation, one due to a velocity approaching speed of light and the other due to the warping of spacetime near great mass densities are actually the same and that the second, described by general relativity, is in fact exactly that: the general description of time dilation.

I'm also drawing on a statement my prof made; As an object approaches the speed of light it becomes more massive. This increased mass and increased gravitational potential would then warp spacetime and explain both the time dilation and length contraction as described by special theory yes?

#### PeterDonis

Mentor
I guess the conclusion I'm hoping to draw is that the two types of time dilation, one due to a velocity approaching speed of light and the other due to the warping of spacetime near great mass densities are actually the same and that the second, described by general relativity, is in fact exactly that: the general description of time dilation.
Sorry, but this is not a valid conclusion. Even on its face it doesn't seem plausible: things can be moving at velocities approaching the speed of light even in empty space far away from any massive objects.

I'm also drawing on a statement my prof made; As an object approaches the speed of light it becomes more massive.
This way of describing what happens is prone to a *lot* of misinterpretation. See further comments below.

This increased mass and increased gravitational potential would then warp spacetime and explain both the time dilation and length contraction as described by special theory yes?
No. An object does not warp spacetime to a greater extent just because it happens to be moving very fast in a particular frame. Warpage of spacetime is frame-invariant. This is a key reason why the statement your prof made is prone to misinterpretation, since it invites the incorrect inference that warpage of spacetime is frame-dependent.

#### arindamsinha

I guess the conclusion I'm hoping to draw is that the two types of time dilation, one due to a velocity approaching speed of light and the other due to the warping of spacetime near great mass densities are actually the same and that the second, described by general relativity, is in fact exactly that: the general description of time dilation...
Note that there are experiments where both types of time dilation show up simultaneously (e.g. Hafele-Keating and GPS satellites). The total time dilation is found to be the sum of the two effects (gravitational time dilation is -ve (i.e. a speeding up) compared to Earth clocks, whereas velocity time dilation is +ve (i.e. a slowing down)). Therefore, they cannot be the same thing.

#### pervect

Staff Emeritus
I guess the conclusion I'm hoping to draw is that the two types of time dilation, one due to a velocity approaching speed of light and the other due to the warping of spacetime near great mass densities are actually the same and that the second, described by general relativity, is in fact exactly that: the general description of time dilation.

Sorry, but this is not a valid conclusion. Even on its face it doesn't seem plausible: things can be moving at velocities approaching the speed of light even in empty space far away from any massive objects.
My \$.02.

While the two certainly are not the same, they are closely related. But related does not mean "the same" of course.

So while I agree with Peter's criticisms in fine detail, on a broader view I think a diffent point needs to be made.

For instance, if one analyzes an accelerating elevator in an inertial frame, and considers a signal sent from the top to the bottom, one finds a velocity dependent doppler shift between the top and bottom of the elevator, due to the transit time.

But one doesn't find any "time dilation" per se, just a doppler shift.

Applying one of the common variants of the principle of equivalence in an accelerating frame, though, one re-interprets the doppler shift as "gravitational time dilation" due to to the pseudo-gravitational field.

I'm not sure how to word this clearly and informally, but basically because "time dilation" is a coordinate dependent concept, it's only possible to clearly say things about it if one talks in great detail about the specific coordinates being used.

The above example clearly (I hope) serves as a specific example of the slipperyness of the concept of time dilation. It's there for one observer, and not for another.

It's very hard to make accurate general statements about time dilation (or any other coordinate dependent concept) at all, because of it's coordinate dependent nature. One runs the common risk of making the statement thinking about it in one specific context which includes a set of coordinates, and then someone else reads it and has in mind a different context, a different set of coordinates, in which the statement is no longer true.

The way round this is pedestrian, and involves explaining the specific context and coordinates used. One still runs the risk of the result being over-generalized . So, let me put in a gratuitous plug for coordinate independence.

It's at best inefficeint to think about physics in terms that dependent on a specific choice of coordinates (like time dilation), because one has to re-invent the wheel everytime one changes coordinates.

Coordinate independent methods are a much better approach - they're easier to talk about accurately, and one doesn't get stuck trying to re-invent the wheel so often.

#### jk22

Yes you could say you use the principle of the aequivalent inertial frame and compute the Lorentz factor but there is a mix up : your escape velocity is classical and you put in a relativistic formula. You should use Ekinetic_relativistic equals Epotential. With Ekinetic_relativistic equals mc*c*(1/sqrt(1-v*v/c*c)-1)

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