accelerating a gravitational field

[bwanaaa]

Newton's equation for gravity calculates gravitational force between two bodies F=GMm/r2
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

γ=1/√(1 − (v^2/c^2))

So, relativistic mass increases with velocity M=mγ The force then increases as gamma squared (because the mass of the planet and the moon both increase)
F=GMm/(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-rela...upport-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.

 

[Gilby]

I found this:
http://www.einsteins-theory-of-rela...upport-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.

There's your error.

Solar systems are not accelerating away from us and reaching a velocity approaching light speed. Rather, space itself is expanding.

 

[videogames101]

Newton's equation for gravity calculates gravitational force between two bodies F=GMm/r2
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

γ=1/√(1 − (v^2/c^2))

So, relativistic mass increases with velocity M=mγ The force then increases as gamma squared (because the mass of the planet and the moon both increase)
F=GMm/(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-rela...upport-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.

You didn't look at the other side of your equation:

F1 = M1a1

F1 = GM1M2 r^2

M1a1 = GM1M2/r^2

a1 = GM2/r^2

a1 = Gm2y/r^2

although it still appears to increase linearly, and I know thats wrong, because switching frames makes it clear the planet and moon would remain in constant orbit with no change in relative force or acceleration.

It would seem however, F= ma will not hold for a non-constant m.

The question that begs is, What is the relativistic version of F = ma? I found one answer.. But I could also be way off base here.

http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html

I'd also keep in mind that a general solution to two bodies moving through space interacting via gravity is not a closed form solution.

http://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity

 

[bwanaaa]

There's your error.

Solar systems are not accelerating away from us and reaching a velocity approaching light speed. Rather, space itself is expanding.

Though I understand the distinction you make, it feels like i am 'cheating' if i say that. Hmmm, The solar systems are separating but they are not moving. Sounds like something a modern day sphinx would ask a modern day Oedipus. This establishes a distinction between
1 ) position and changes in position in the universe and
2 ) movement through the universe.

But both processes - 1) change in position due to spatial expansion 2) change in location due to movement - are measured the same way. And in fact relativistic effects are seen for stars that are separating from us due to spatial expansion. So why would relativistic changes in mass not be expected?

 

[bwanaaa]

It would seem however, F= ma will not hold for a non-constant m.

The question that begs is, What is the relativistic version of F = ma? I found one answer.. But I could also be way off base here.

http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html

Thank you but I think he could be clearer.
That author says that force is not always parallel to the acceleration. I dont get what he means. Anyway he goes on to use vectors to derive:

F = (I + γ2 v vt) γ m a
Where vt is the velocity as a row vector, and I is the 3 × 3 identity matrix.
and in the next line he says
a = (1 – v vt) F
—————
γ m

He concludes with 'The directional dependence that the newtonian meaning of mass has now taken on is neatly contained in the matrices 1 + γ2 v vt and 1 – v vt, and the remaining factor γ m is the relativistic mass. '

I dont get the omitted details of his approach-in particular why he complexifies the solution and I wonder if he made a confusing typo as well when he says 'As usual, set γ = (1 – v2) –1/2.' That's not gamma as it is usually defined.
He seems to be making a distinction that mass will be different if you measure it in one direction versus another. Ok, that predicts that the orbital mechanics of a solar system would appear even weirder- the gravitational force on the planet will change as it orbits its sun.

 

[bwanaaa]

You didn't look at the other side of your equation:


I'd also keep in mind that a general solution to two bodies moving through space interacting via gravity is not a closed form solution.

http://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity

Tnx again for finding that. You make me look lazy as if I did not search - but it's just that I did not know to state the question using appropriate physics jargon in google or wikipedia. The end of the article goes on to show that numerical methods have been used to calculate that gravitational field for two bodies orbiting each other. The point is that two massive bodies will emit gravitational radiation and lose energy and spiral in towards each other. The velocity of these bodies through the universe is not addressed. Still, the increased relativistic mass as a result of their motion through the universe would make them behave as truly massive objects and cause them to emit gravitational waves.

 

[bwanaaa]

The observations of Vera Rubin and others regarding the orbital mechanics of far away galaxies is a concrete example of the problem I am describing. And in her case, the data shows that galaxies are spinning a lot faster than they should be. This has been interpreted as the evidence for dark matter. I am wondering if some of these 'aberrant' orbital mechanics can simply be explained by relativistic increases in mass . I am sure the people who 'invented' the idea of dark matter thought of this but I cannot find evidence of their calculations. I guess you cannot get published for 'non-discoveries'. And newly invented words go a long way when you have to write a grant.

 

[DominionSeraph]

The observations of Vera Rubin and others regarding the orbital mechanics of far away galaxies is a concrete example of the problem I am describing. And in her case, the data shows that galaxies are spinning a lot faster than they should be. This has been interpreted as the evidence for dark matter. I am wondering if some of these 'aberrant' orbital mechanics can simply be explained by relativistic increases in mass . I am sure the people who 'invented' the idea of dark matter thought of this but I cannot find evidence of their calculations. I guess you cannot get published for 'non-discoveries'. And newly invented words go a long way when you have to write a grant.

Riiiiiiiight.
You have no idea how science works.

 

[chimaxi83]

Riiiiiiiight.
You have no idea how science works.

 

[Biftheunderstudy]

This may be the conglomeration of some slight misconceptions.

First, relativistic mass is not a thing, it only looks and behaves that way to the outside observer. The 2 orbiting bodies being in the same reference frame, only the relative velocity difference really matters.

But more to the point (and this was already said), galaxies aren't moving through space according to the Hubble relation. The recessional speed is a misnomer, it is the space that is increasing between them. As such, those galaxies stars and planets aren't moving at the speed of light.

We can see this by looking at the CMB and measuring our galaxies velocity relative to it, I believe it comes out to somewhere around 700 km/s, way smaller than c.

Now, if they actually were moving relativistically, there would indeed be some weird orbital mechanics at play (the non parallel forces and such). And it wouldn't be the first time if we forgot about relativity when looking at distant objects. An example would be the relativistic jets that distant quasars emit, it was first thought that they were super-luminal. But upon closer examination, and remembering some kind of counter intuitive relativity it turned out to be a projection effect.