Objects accellerating due to each other's gravity

[michelcolman]

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.

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

 

[mfb]

No, that does not work. I guess someone will post a detailed explanation about this soon.

I basically just wanted to get an idea of the order of magnitude of accellerations depending on how fast the objects were going.

Well, in the limit of zero speed and large distances, you get the right (Newtonian) result, so if the speed is not too large and the distance is not too close, the result will be in the same order of magnitude of the real value.

 

[D H]

As mfb noted, your approach isn't valid.

One (somewhat) valid approach is to use a parameterized post-Newtonian (PPN) approximation of general relativity. These approaches keep time as the independent variable and model relativistic effects as perturbations of Newtonian gravity. Like Newtonian gravity, these PPN approximations are just that, approximations. They are approximately valid if relative velocities between objects are small compared to the speed of light and if relative distances between objects are large compared to the Schwarzschild radii.

In addition to modifying how you compute gravitational acceleration you'll also need to modify how you represent time. Even with a PPN approximation you have to go all-in or you're going to get worse results than you would have just using good old Newtonian physics.

Here's a link that describes one such PPN approach: http://tai.bipm.org/iers/conv2010/conv2010.html. Pay particular attention to chapter 10.

 

[michelcolman]

I was interested precisely in the case of high relative speeds approaching the speed of light, so I suppose those PPN approximations won't do the trick.

But surely, for my extremely simple one-dimensional example of two masses m1 and m2 at speeds v1 and v2 and distance d1 and d2, there must be some way of calculating their accellerations? Is GR so complicated that even this simple example requires enormous amounts of math and computer simulations?

 

[Mentz114]

But surely, for my extremely simple one-dimensional example of two masses m1 and m2 at speeds v1 and v2 and distance d1 and d2, there must be some way of calculating their accellerations? Is GR so complicated that even this simple example requires enormous amounts of math and computer simulations?

Yes. No simple (ie closed) solution of the EFE has been found for this scenario.

 

[michelcolman]

Not even for the immediate accelleration in one dimension? I can understand that the full two-body threedimensional motion equation (calculating entire orbits) may be rather complicated, but my example seemed to be so extremely simple, and I'm asking for so little, that it's hard to believe that this would currently be impossible to solve.

 

[D H]

But surely, for my extremely simple one-dimensional example of two masses m1 and m2 at speeds v1 and v2 and distance d1 and d2, there must be some way of calculating their accellerations? Is GR so complicated that even this simple example requires enormous amounts of math and computer simulations?

The answers to your questions are no and yes, respectively.

Each of the quantities you are talking about, mass, velocity, and distance, are a bit problematic in general relativity. This is particularly so with regard to velocity and distance. By saying "at speeds v1 and v2 and distance d1 and d2" you are implicitly assuming there exists some global reference system such that one can uniquely and meaningfully ascribe some set of space-time coordinates to an event in space and time. Taking time derivatives to yield velocities just compounds the issue.

That global reference system simply does not exist in general relativity. Space-time becomes highly non-Euclidean in special relativity at high velocities. The problem is compounded in general relativity. Coordinate systems are local (not global) charts on the non-Euclidean space-time manifold. These issues can be hand-waved away when masses are small, velocities are low. Those hand-waves don't work in the case of large masses, high velocities, and short distances -- and that is precisely the region you want to investigate.

 

[Mentz114]

Not even for the immediate accelleration in one dimension? I can understand that the full two-body threedimensional motion equation (calculating entire orbits) may be rather complicated, but my example seemed to be so extremely simple, and I'm asking for so little, that it's hard to believe that this would currently be impossible to solve.

I did not say it was impossible, just that it has not been found. Of course, it probably is impossible, but we can't even say that with certainty.

 

[michelcolman]

The answers to your questions are no and yes, respectively.
Each of the quantities you are talking about, mass, velocity, and distance, are a bit problematic in general relativity. This is particularly so with regard to velocity and distance. By saying "at speeds v1 and v2 and distance d1 and d2" you are implicitly assuming there exists some global reference system such that one can uniquely and meaningfully ascribe some set of space-time coordinates to an event in space and time.

Not at all, I am well aware that there's no such thing as a single global reference system. But one observer can pick his own reference system and apply the laws of physics in that reference system. He will make different measurements than someone else who is using a different reference system, and they can get wildly different results for distances, speeds and times, but any conclusions about actual events will match. At least that's what I understood from relativity.

So I just wanted to know the result from the point of view of one particular observer who observes these masses with rest mass m1 and m2, distance d1 and d2, and speed v1 and v2, using his "measuring sticks", his clock, his planes of simulaneity, etcetera.

Space-time becomes highly non-Euclidean in special relativity at high velocities. The problem is compounded in general relativity. Coordinate systems are local (not global) charts on the non-Euclidean space-time manifold. These issues can be hand-waved away when masses are small, velocities are low. Those hand-waves don't work in the case of large masses, high velocities, and short distances -- and that is precisely the region you want to investigate.

I knew it was difficult in general, what with frame dragging effects and things like that, but I was sort of hoping that the situation for two point masses flying in the same direction would be doable. Apparently I was wrong.

 

[pervect]

So I just wanted to know the result from the point of view of one particular observer who observes these masses with rest mass m1 and m2, distance d1 and d2, and speed v1 and v2, using his "measuring sticks", his clock, his planes of simulaneity, etcetera.

I can provide a reference to the literature that computes something that's closely related and avoids a lot of sticky definitional issues.

Consider a cloud of test particles, all at rest relative to each other. Then suppose a massive object flies through them at ultrarelativistic velocities.

Trying to measure and specify the changes in relative velocites instant by instant turns out to be extremely difficult. Specifying the acceleration of each test object pretty much requires a global coordinate system, the thing that you agreed was problematic. And very messy, to boot.

Looking at the relative acceleration of nearby test objects turns out to be another way to go, but when I try to present this approach, people rarely seem to think it answers their questions. (It boils down to considering tidal gravity and leads to a discussion of the Riemann tensor and geodesic deviation.

But there is a third thing you can do.

What can be done relatively easily is to calculate the velocity field of the test particles before and after the flyby. (In fact, oe sets the velocity field to zero before the flyby). And compare the results to what happens with Newtonian theory.

This is done by Olson in http://dx.doi.org/10.1119/1.14280, "Measuring the active gravitational mass of a moving object ".

I'll quote from the abstract. I believe the whole paper itself was available somewhere (if you can find it, it's worth a read).

f a heavy object with rest mass M moves past you with a velocity comparable to the speed of light, you will be attracted gravitationally towards its path as though it had an increased mass. If the relativistic increase in active gravitational mass is measured by the transverse (and longitudinal) velocities which such a moving mass induces in test particles initially at rest near its path, then we find, with this definition, that Mrel=γ(1+β^2)M. Therefore, in the ultrarelativistic limit, the active gravitational mass of a moving body, measured in this way, is not γM but is approximately 2γM.

THis presents a clearcut and easily interpretable view of the gravitational effect of a massive object in terms of things you can actually measure . It's integrated over time in terms of the total impulse delivered, but it can easily be seen that the answers that GR predicts do depend on energy, but they don't depend on energy in a linear manner. There's an additonal factor of (1+beta^2) that mere "futzing around" with Newtonain theory doesn't predict.

 

[michelcolman]

But there is a third thing you can do.

What can be done relatively easily is to calculate the velocity field of the test particles before and after the flyby. (In fact, oe sets the velocity field to zero before the flyby). And compare the results to what happens with Newtonian theory.

This is done by Olson in http://dx.doi.org/10.1119/1.14280, "Measuring the active gravitational mass of a moving object ".

I'll quote from the abstract. I believe the whole paper itself was available somewhere (if you can find it, it's worth a read).

Too bad it's behind a paywall. But of course this is one massive object and a bunch of massless test particles, which is a bit simpler than two massive objects. Still, it would be interesting to read the whole thing if it's not too technical.

 

[D H]

Google the title. You'll find the article. I can't post a link because doing so might violate copyright law.

 

[mfb]

Mrel=γ(1+β^2)M. Therefore, in the ultrarelativistic limit, the active gravitational mass of a moving body, measured in this way, is not γM but is approximately 2γM.

An additional reason to avoid "relativistic masses", thanks.

 

[Bill_K]

We discuss this issue often. In the most recent thread I offered this remark about the "factor of 2" in the Olson-Guarino paper, comparing it to the reputed factor of 2 between the Newtonian and GR values for light deflection:

Yes, I think this is the same factor of 2. It's not really the difference between Newton and GR, it's the difference between scalar gravity and tensor gravity. A particle sitting still feels only the scalar (Newtonian) potential. A moving particle feels also the vector component (~v) and the tensor component (~v²). Hence the 1 + β² factor.

 

[TheEtherWind]

How about

a1(t)=GM/[x1(t)-x2(t)]^2

a2(t)=-GM/[x1(t)-x2(t)]^2

These are the differential equation explaining two objects falling toward each other. In a straight path of course.

Edit: And of course assuming Newtonian gravity suffices as an approximation.