Does Relativistic Time Dialation effect Gravity?

[Iron Sun 254]

My question is simply this: Since the calculation of gravitational force includes a time element, shouldn't this force be reduced because of the time dialation which occurs due to relativity?

[Chris Peterson]

My question is simply this: Since the calculation of gravitational force includes a time element, shouldn't this force be reduced because of the time dialation which occurs due to relativity?

If you are considering gravity as a force, you are presumably looking at it from a Newtonian standpoint. As such, the only place time enters the equation is inside the universal gravitational constant. It is understood that a strictly Newtonian analysis deviates from reality if the system is in a relativistic regime.

If you are considering gravity from the viewpoint of GR, the effects of time dilation are inherent in the mathematical description.

[Iron Sun 254]

Actually, I'm talking about Special Relativity. My question is related to the gravitational force experienced by one body by another body based on the relative velocity of the two.

Here's the thought experiment that got me to this point and then we can go from there. Imagine if you will a body identical to the Earth comes zipping through the Solar System at .9c. Now, this planet is identical to the Earth in every way, or so it seems... because if I watch Little Johnny bouncing his basketball on that other world, what I'd notice is that it takes longer for that basketball to reach the ground. Also, if I time how long it takes that other world's moon to orbit, it takes longer to complete a full orbit, in fact, it'll take more than twice as long as it takes our moon to orbit Earth. Based on all of this, it would seem that the gravitational force of that other world is less than our own.

Now, to take it one step further, let's say this world passed by our own Earth at a point where our moon was equidistant between our world and theirs. Our moon shouldn't experience a stronger pull towards that other world than it's own moon does. They should, in fact be the same, and, based on our frame of reference, the pull of the other world would seem to be less than ours.

In the same way that time dialation, length contraction, etc. can be the only way to reconcile the speed of light between two frames of reference, a reduction of the gravity between different frames of reference seems to be the only way to reconcile the observed behavior.

[Chris Peterson]

Actually, I'm talking about Special Relativity. My question is related to the gravitational force experienced by one body by another body based on the relative velocity of the two.

SR doesn't deal with gravity. But GR does include time dilation and other effects that occur in observations between different non-inertial frames.

So I think your thought experiment fails because you aren't analyzing it in the proper regime, which is GR, not SR.

[alter-ego]

Actually, I'm talking about Special Relativity. My question is related to the gravitational force experienced by one body by another body based on the relative velocity of the two.

As Chris suggested, your thought experiment is incorrectly presented. You basic question is a good one, and special relativity does provide insight towards a conclusion, but you have misinterpreted what SR is telling you. Let's start by frankly stating that SR does not explicitly deal with gravity. However, with constraints, you can deal with acceleration in SR:

Can Special Relativity handle accelerations?
It is a common misconception that Special Relativity cannot handle accelerating objects or accelerating reference frames. It is claimed that general relativity is required because special relativity only applies to inertial frames. This is not true. Special relativity treats accelerating frames differently from inertial frames but can still deal with them. Accelerating objects can be dealt with without even calling upon accelerating frames. ...

A good example is Einstein first described gravitational time dilation by incorporating acceleration into SR.

Now take your example of two identical planets passing by each other at a known constant velocity. You are correct in that event timing measurements made by planet 1 of events on planet 2 will reveal SR time dilation, but applying that obervation to modifying gravity on the other planet is incorrect. Concluding that clocks on planet 2 are running slower is the correct SR interpretation. Nothing more, nothing less. In fact, a modified thought experiment will lead you to conclude that the mass is the same. The time dilation (slower clocks) cancels the false conclusion that planet 2 gravity / mass is less.

OK, here's the kicker, and SR is the source of this interpretive extrapolation: SR predicts that the relativistic mass (energy!) increases with velocity - Like time dilation, this is a fundamental result. Well, if your first thought is that gravity should increase, you would be right!! The details are complicated, but the interpretation is not. From GR, gravity (space-time curvature) originates from all forms energy, kinetic, thermal, radiation, mass (=mc^2). Therefore, the real answer about gravity and motion is that planet 2 "gravity" increases with velocity, and detailed GR tests and calculations will reveal that effect to observers on planet 1.

So Iron Sun, it appears to me you were mislead by incorrectly applying SR and time dilation to conclude that gravity should change. But, your thought about SR telling you something, though indirectly, about gravity is valid. It's not time dilation though, but mass "dilation". I say this because mass/energy is scaled by the same factor, γ.