# accelerating a gravitational field

Newton's equation for gravity calculates gravitational force between two bodies $F= GMm/r^2$

My question is: How does the force change as these two bodies accelerate together through the universe and approach light speed?

You might argue there should be no change because both bodies are accelerating together and stay the same relative to each other. Consider a planet and a moonbase. They people on each site will experience the same time dilation according to special relativity. Special relativity allows us to adjust the kinematics equation for an object as it approaches the speed of light. The Lorentz factor concisely describes this

$\gamma=1/\sqrt{1 \ - \ v^2/c^2}$.

So, relativistic mass increases with velocity $M=m\gamma$ The force then increases as gamma squared (because the mass of the planet and the moon both increase) $F= G \frac{Mm}{ (1 \ - \ v^2/c^2) \ r^2}$. If I read that right, it says force goes to infinity as a planetary pair reach lightspeed together.

I found this: http://www.einsteins-theory-of-relativity-4engineers.com/support-files/velocity-effects-on-gravityY.pdf But it seems not exactly relevant as it discusses the measurement of gravity between two masses moving at relativistic speeds to each other. I am talking about the gravity within a solar system that is accelerating and reaching a velocity away from ours approaching light speed.

If gravitational force indeed increases then orbital mechanics of the solar system would be different than ours. Solar systems very far away (which happen to be close to light speed due to the Hubble expansion) would have a much greater gravitational milieu. That means that the planets orbit their sun a lot faster than expected at a given distance from their sun.

Please point out the error in my analysis gently. This discussion assumes the gravitational constant,G, does not change as objects accelerate and reach light speed.

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Aquagremlin - can you be a bit more specific about the scenario you consider? It seems you wonder about two massive objects traveling together as seen by a distant observer who moves at relativistic speed compared to the two bodies? However, you also mention "people on each site [on both bodies] will experience the same time dilation according to special relativity". – Johannes Jul 28 '13 at 5:58
"You might argue there should be no change because both bodies are accelerating together and stay the same relative to each other" you are misunderstanding something here for sure. – TMS Jul 28 '13 at 8:55
@Johannes, THANK YOU for making my post look readable! Can you tell me how you did it? As for clarification, I think you understood it well. Yes, this is what I meant " It seems you wonder about two massive objects traveling together as seen by a distant observer who moves at relativistic speed compared to the two bodies?". Leave out the bit regarding time dilation, it is not really germane to the question. I just wanted to point out that a distant observer and the set of two massive bodies are separating at relativistic speeds,but that the two massive bodies are staying together. – aquagremlin Jul 28 '13 at 12:21
@TMS Can you be more specific about your objection? It is not helpful. One might argue that you misunderstood the question. – aquagremlin Jul 28 '13 at 12:23
@Aquagremlin - you're welcome. When new on this site, like you, I had no clue about how to render equations. Others edited the equations in my posts, and I learned simply by clicking 'edit' and looking at the equation codes. It's really simple. Start with a dollar sign and end with a dollar sign (double dollars for non in-line equations) and off you go... – Johannes Jul 28 '13 at 13:50