# Newton's Gravity & Lorentz Contraction: Is Modification Needed?

• I
• Kashmir
It's not clear what you are asking. If you are asking whether length contraction by itself is enough to determine that the Newtonian gravitation law must be modified, the answer is no. If you are asking how close a result length contraction can get us to the right answer, the answer is sometimes. But it is not clear how we could tell you what "close enough" is. In summary, the answer to your question is no.

#### Kashmir

Newton's gravity depends on the euclidean distance between two masses.
Two comoving frames will have different values of length between masses so the forces will be different in two frames.
Is it enough to prove that the gravity rule has to be modified?

Lluis Olle, Kashmir and topsquark
That's a very misleading homework. If they had ask for the motion of a particle in the Coulomb field of a fixed (i.e., very massive) particle, neglecting radiation reaction, it'll have been fine, but to think one could use the Newtonian gravitational force in relativity is at least a bit questionable. One should rather ask for the solution of the geodesic worldline in the Newtonian approximation of the Schwarzschild spacetime, if one wants to discuss the motion of a planet around the Sun.

Orodruin and topsquark
PeroK said:
Here's an interesting homework assignment
I'm not sure I see how this thread is helpful. The thread left it hanging whether the answer given was correct (I don't think it is); in fact the thread left hanging a highly pertinent question from @Orodruin, namely, how ##\gamma## is to be defined, or more precisely how the speed ##v## is to be defined; is it the actually measured speed relative to static observers, or just the coordinate speed?

If there is a proof somewhere in the literature that, at least for some special cases, the heuristic "plug the Newtonian gravity formula into SR" happens to work, a reference to that might be helpful. I'm not sure any such reference exists. But certainly the thread referred to is not such a reference.

vanhees71
Kashmir said:
Newton's gravity depends on the euclidean distance between two masses.
Two comoving frames will have different values of length between masses so the forces will be different in two frames.
Is it enough to prove that the gravity rule has to be modified?
There is a related question to this, namely, that Newton's gravity depends on the relative positions of the masses at some instant of time. But "instant of time", i.e., simultaneity, is frame dependent in relativity; if the two masses are moving relative to each other, their notions of simultaneity will be different, and which one should be used for the Newtonian force law? Is that enough to prove that the gravity rule has to be modified?

Einstein explicitly considered the latter question in 1907, and his answer was yes. I don't know if he explicitly considered whether the frame dependence of the distance itself would lead to a similar answer, but I think it would.

vanhees71
Kashmir said:
Newton's gravity depends on the euclidean distance between two masses.
Two comoving frames will have different values of length between masses
PeterDonis said:
Newton's gravity depends on the relative positions of the masses at some instant of time. But "instant of time", i.e., simultaneity, is frame dependent in relativity
PeterDonis said:
Einstein explicitly considered the latter question in 1907
Actually, I think the question Einstein considered in 1907 was more basic than either of the above: Newtonian gravity says that gravity propagates instantaneously, but "instantaneously" is frame-dependent in relativity. A relativistic theory cannot have instantaneous propagation; instead, one would expect that a relativistic interaction would propagate at the speed of light, as electromagnetism does. But just plugging a speed of light interaction speed into Newtonian gravity does not work. Carlip's classic paper on aberration and the speed of gravity discusses this:

https://arxiv.org/abs/gr-qc/9909087

vanhees71
PeterDonis said:
but I think it would.
You think its enough , this length contraction to decide that Newtons gravitation should be modified?

Kashmir said:
You think its enough , this length contraction to decide that Newtons gravitation should be modified?
Does it matter? You know the answer - that GR is needed - and in a sense that is the only answer. Asking whether length contraction by itself is enough is somewhat hypothetical.

Kashmir said:
You think its enough , this length contraction to decide that Newtons gravitation should be modified?
Focusing on length contraction specifically is not really correct. Length contraction is not the only relevant consequence of SR that affects the Newtonian gravitation law, as I show in posts #6 and #7. So the correct question to ask is simply whether the Newtonian gravitation law can be incorporated as-is into a theory based on SR. And the answer to that question is no.

Will this give you the right answer? No.
Will this give you an answer that is close? Sometimes.
Is it close enough? How could we possible tell you what "close enough" is?

PeroK said:
That is not interesting. It is fundamentally flawed … as I pointed out in that thread …

vanhees71 said:
That's a very misleading homework. If they had ask for the motion of a particle in the Coulomb field of a fixed (i.e., very massive) particle, neglecting radiation reaction, it'll have been fine
As stated
Orodruin said:
It is particularly unfortunate as you could actually make a reasonable problem by replacing gravity by the motion of a test charge in a static electric field.

PeroK said:
Does it matter? You know the answer - that GR is needed - and in a sense that is the only answer. Asking whether length contraction by itself is enough is somewhat hypothetical.
I know that GR is needed, I was just curious and tried to see how many things went wrong with Newtonian mechanics in SR setting.

Kashmir said:
I know that GR is needed, I was just curious and tried to see how many things went wrong with Newtonian mechanics in SR setting.
Newtonian mechanics and SR is pretty similar. There are equivalents of all three of Newton’s laws. The big difference being the structure of spacetime.

Newtonian gravity on the other hand is fundamentally incompatible with the spacetime structure of special relativity.

vanhees71
Kashmir said:
how many things went wrong with Newtonian mechanics in SR setting.
Just one. You get the wrong answer.

But isn't that enough?

vanhees71