# Newtonian gravity and Special Relativity

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1. Jun 30, 2014

### analyst5

Hey guys,

in classical mechanics Newton's law of universal gravitationa force says that the force between two bodies is equal to the product of their masses divided by the square root of the distance between them. So far, so good. In SR, lengths depend on the frame that we are in, and so the parameter 'r' (distance) varies between frame to frame and therefore that implies that the gravitational force varies from frame to frame. This fact should also imply that the acceleration which a body undergoes because of gravity varies from frame to frame. Is this valid reasoning, and could somebody help me with the maths? The basic question is can Newton's law of gravity be compatible with SR.

Regards, analyst.

2. Jun 30, 2014

### WannabeNewton

You know, general relativity exists for a reason...

The answer to your question is "no"; Newton's law of gravitation, or more generally Poisson's equation for the gravitational potential, $\nabla^2 \varphi = 4\pi \rho$, has to be augmented in order to be compatible with SR. It is not a Lorentz covariant equation which is the main issue. Compare it with Maxwell's equations which are Lorentz covariant.

3. Jun 30, 2014

### Staff: Mentor

More or less, but there's also another problem: Newtonian gravity is instantaneous--the force at a given point depends on the mass and radius at the same time. But in relativity, "at the same time" is frame-dependent. This was the main thing that led Einstein to conclude that Newtonian gravity was not compatible with SR.

Not as it stands, no. People have proposed ways to modify Newtonian gravity to make it compatible with SR, without going all the way to General Relativity; but all other proposed theories except GR have been falsified by experiments.

4. Jun 30, 2014

### analyst5

Ok, I get the instantaneous propagation problem, but could you give me a mathematical clue on how would the acceleration due to gravity vary between frames due to length contraction, if we use the original Newton formula for gravity and his second law of motion and combine it with relativistic effects?

5. Jun 30, 2014

### Staff: Mentor

No; the correct response is "don't do that". Seriously, you can't combine relativistic effects with something that explicitly violates relativity, like Newton's original laws of motion and gravity; you'll get nonsense.

6. Jun 30, 2014

### analyst5

Ok, can you at least give me (if it's not a problem) some references or insights on relativistic reworks of Newtonian gravity, based on SR so I can just get a clue?

7. Jun 30, 2014

### Staff: Mentor

You mean, other than GR? After all, GR itself can be viewed as a "relativistic rework of Newtonian gravity, based on SR", since it is based on the equivalence principle, which says that anywhere in spacetime you can always find a local inertial frame in which the laws of SR hold good, and in which "gravity" can't be detected at all.

As for other attempts to formulate a theory of gravity within a relativistic framework, you might start with Brans-Dicke theory, which tries to model gravity using a scalar field as well as a metric tensor, and the "spin-2 field on flat spacetime" theory, which started as an attempt to find a quantum theory of the "graviton", the quantum particle that would mediate the gravitational interaction (just as all other interactions have particles that mediate them), but ended up showing that the classical limit of such a theory is in fact GR itself. These links will get you started:

http://en.wikipedia.org/wiki/Brans–Dicke_theory

http://en.wikipedia.org/wiki/Graviton

It's worth noting, though, that *none* of these theories take the form of "start with Newtonian gravity and then modify it to fit in a relativistic framework", because, as I said before, that doesn't work. So if that's what you mean by "a relativistic rework of Newtonian gravity", then there is no such thing. For all these relativistic gravity theories (including GR), Newtonian gravity only serves as a check on the theory once it's been developed by other means, i.e., it should be possible to show that the laws of the theory reduce to Newton's laws in the appropriate approximation (basically, when gravity is weak and everything is moving very slowly compared to the speed of light). But this reduction is one-way; you can't go the other way and get the full correct theory from Newton's laws by "expanding" them to cover cases where gravity is not weak or objects are moving at relativistic velocities.

8. Jun 30, 2014

### DrStupid

There are two possible ways to do that:

1. Transform the positions from one frame of reference to the other and calculate the gravitational force using the result.

2. Calculate the force in one frame of reference and transform it to the other.

With Galilean transformation you will get the same results in both cases. With Lorentz transformation the results will usually be different.

9. Jun 30, 2014

### Vorpal

To do this, you would have to break the principles of special relativity by introducing an absolute frame. One can generalize this slightly by introducing a scalar field t that serves as the absolute time of an event. Then the resulting Newtonian-SR gravitational acceleration on a particle with four-velocity $u$ is:
$$a_\text{g} = G\int\frac{T^{\alpha\beta}t_{,\alpha}t_{,\beta}}{(l\cdot l)^{3/2}}\left[l+(u\cdot l)l\right]\,\mathrm{d}^3x'\text{,}$$
where $l$ is the spacelike vector connecting $x'$ to $x$ and $\mathrm{d}^3x'$ is the volume element on the constant-t slice. The basic idea is that only the $T^{00}$ energy density term contributes in the absolute frame, and the $(u\cdot l)l$ projection corrects for the case where the particle is not at rest in the absolute frame. (Ref: Lightman et al's Problem Book in Relativity and Gravitation.)

Note that the Newtonian gravity would actually use the rest mass density, but here we use the (mass-)energy density instead. This is another casualty of forcing the Newtonian inverse-square law into special relativity, in addition to the main one of forcing us to introduce an absolute frame.

Newtonian gravity is dead. It's just wrong. However, there is a general scheme for comparing a fairly large class of gravitational theories called the parametrized post-Newtonian formalism in the regime of weak gravitational fields. You can see a large collection of various gravitational theories you can look up through the list of their PPN parameters here.Not all theories of gravity would fall neatly into the PPN scheme, but it's a start on some alternatives.

10. Jun 30, 2014

### Staff: Mentor

I think that the closest thing that you will find is Nordstroms theory. As has been mentioned already, it is falsified by experiment.

http://en.m.wikipedia.org/wiki/Nordström's_theory_of_gravitation

11. Jul 1, 2014

### haushofer

The reason why nordstroms theory fails with experiment is e.g. because a grav. scalar field would couple to the trace of the energy momentum tensor. For an el.magn.field this trace is zero, hence no deflection of light by gravity. See e.g. Ortin's Gravity and Strings.