Is there a (relatively) simple formula for calculating the accelleration of objects due to each other's gravity, but taking relativity into account? Can I just adjust Newton's law of gravity, or do I need to dive into (and probably drown in) general relativity? I would just like some formula that is functionally equivalent to Newton's law: given the positions, masses and speeds of two objects, calculate their accellerations from the point of view of a particular observer.(adsbygoogle = window.adsbygoogle || []).push({});

Here's what I came up with so far, for the following simple scenario:

Imagine two objects flying directly away from us in the same direction. The furthest object has rest mass m1, speed v1, and is at a distance d1. The other object of course has m2, v2 and d2 with d2 < d1. Imagine they are far enough away from us for their gravitational potential not to have any effect on our own clocks.

Newton's law gives a force of G m1 m2 / (d1-d2)^2

The first thing I'd do is replace m1 and m2 with the relativistic masses which are higher due to v1 and v2.

Then, (d1-d2) does not take into account the fact that gravity travels at the speed of light. I should use the distance the objects were at when the gravitational "signal" left them.

For m1, this distance becomes (d1-d2)*c/(c-v2) (which is more, resulting in less force on m1 from m2)

And for m2, (d1-d2)*c/(c+v1) (which is less, resulting in more force on m2 from m1)

At first sight it would seem strange to have different forces acting on the two objects, but I guess that's just one of those relativistic paradoxes that disappears when you actually calculate some observable result. Some other observer may get the opposite result for the forces (m1 getting more force instead of less) but when it comes to calculating collisions or things like that, they'll still arrive at the same outcome. Maybe the same amount of energy is being delivered to both objects over a different amount of time (depending on differing planes of simultaneity) which would explain the differing forces.

Anyway, after these corrections on mass and distance, I would apply the relativistic version of F=m*a (I know, it probably looks like a horrible formula with square roots and not at all linear, but surely you can still convert a force to an accelleration somehow).

Would this be anywhere close to the correct result? Or am i completely misguided in trying to use an obsolete formula for something like this? If I am, what's the alternative? (Without pages full of tensor math). From what I've seen so far, GR is usually about the effect of some massive object on its surroundings (distorting space-time), but here we have two objects affecting each other.

I basically just wanted to get an idea of the order of magnitude of accellerations depending on how fast the objects were going. I was doing thought experiments with spaceships flying towards stars (or the stars flying towards them from their point of view) and what kind of measurements different observers would be making, but it was getting too complicated without some formula to hang on to.

Thank you,

Michel Colman

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# Objects accellerating due to each other's gravity

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