# Few concepts that I miss in SR

1. Jan 2, 2009

### Heimdall

Hi,

Could you explain me a few things I still don't get about special relativity ?

I've been tought a few years ago that special relativity wasn't good for handling accelerations... well I've been reading books and a few topics here, and I'm now convinced it is not true... for exactly the same reason galilean mechanics can handle accelerations as seen from inertial frames...

ok....

My question is... As SR can handle accelerations (forces), I can calculate what is, in an inertial reference frame, the trajectory of a relativistic particle in an electromagnetic field. To do this, I have to write the relativistic equation of the dynamic :

$$\frac{d\mathbf{P}}{dt} = \mathbf{F}$$

where $$\mathbf{P}$$ and $$\mathbf{F}$$ are the 4-vector energy-impulsion and 4-force respectively.

$$\mathbf{F}$$ in this problem, being the electromagnetic Lorentz-force.

It is often said "SR is good at handling accelerations, as long as there is no gravity", that I don't understand !

Why couldn't I replace the electromagnetic field in the previous problem, by a gravitationnal field (newtonian GM/r^2) ? Why couldn't I solve the dynamic of a relativistic particle in the gravitationnal field of the sun for example ?

I've been told many things which have not convinced me. Among them :

- the reason would be that electromagnetic interaction is possible in SR because it propagates at a finite speed whereas newtonian universal gravitation violates the principle of relativity by propagating at a infinite speed. I accept these facts, but I can't see why it would explain that one can use lorentz force but not gravitational force in SR. (after all there are probably some cases where the body has a relativistic velocity but the propagation time of the gravity is negligeable ?)

- the reason would also be that electric charge is invariant in respect to the inertial transformation, whereas the mass is not. I don't even understand that... to me, the mass is also conserved ... no ? It is just the inertial factor (gamma times the mass) that changes. How could this explain that gravitational forces are prohibited in SR ?

Further related question... if electromagnetic forces are tolerated, how can a electromagnetic two body problem can be formulated ? If I take an electron and a proton, moving at some relative speed, they will feel attraction due to coulomb force, and their trajectory will be somehow changed. From an inertial reference frame, their motion can, according to me, be described via the relativistic equation of the dynamic... where the force is the coulomb force.

What I don't understand in what I've just said, is that with coulomb force, there is no time in my equations, so any change in the electron motion will immediatly be felt by the proton... just like in Newtonian physics (and newtonian gravitational force) thus violating the principle of relativity... exactly as if I would have written the gravitationnal two body problem with the newton universal gravitation law..

Where is my mistake ?

2. Jan 2, 2009

### JesseM

If you assume the Newtonian force equation works in some specified inertial frame, I think the answer is that you could then solve the dynamics of particles, the problem would be that if you transformed into other inertial frames the force equation would have to look different in order to avoid conflicting predictions--Newtonian gravity is not "Lorentz-symmetric" like electromagnetism is. And if there are any laws of physics that don't follow the same equations in all inertial frames, this is a violation of the first postulate of SR.
It's not the mass that fails to be invariant, it's the dynamical equations of the theory that won't remain the same if you transform these equations into a new frame using substitutions like x=gamma*(x' + vt') and t=gamma*(t' + vx'/c^2). Consider the equation for the acceleration a particle feels from an object of mass M fixed at coordinates (x0,y0,z0)--in this case the acceleration experienced by a particle at coordinates (x,y,z) would be GM/r = $$\frac{GM}{\sqrt{(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2}}$$. Now if you use the Galilei transformation you have x = x' + vt', y = y', z = z', and also x0 = x0' + vt', y0 = y0', z = z0'. If you substituted these values into that equation for the acceleration, it gives you back exactly the same equation but with unprimed variables replaced with primes: $$\frac{GM}{\sqrt{(x' - x'_0)^2 + (y' - y'_0)^2 + (z' - z'_0)^2}}$$. On the other hand, if you were to do a substitution using the Lorentz transformation equations like x = gamma*(x' + vt') then the new equation would look quite different...the Newtonian gravity equation is Galilei-symmetric but not Lorentz-symmetric.
The coulomb force equation only works in the case of a source that's at rest in the frame you're using and is stuck at the same fixed position, the force between moving particles is not given by the Coulomb force, you have to use the full set of Maxwell's Laws to figure out their dynamics, and if you do you find that distant particles act like they aren't "aware" of the acceleration of a source until an electromagnetic wave moving at the speed of light (created at the position and time the source accelerated) reaches them. See the section on this page titled "Radiation as a Consequence of the Cosmic Speed Limit" (the previous sections also give some good info on how the same laws of electromagnetism give consistent predictions in different relativistic frames).

3. Jan 2, 2009

### Heimdall

So if I understand well, Newtonnian et Galilean physics was ok for gravitation but not for electromagnetism... and special relativity is ok for electromagnetism but not anymore for gravitation...

I see your point, but isn't electromagnetism built from the coulomb force ?

4. Jan 2, 2009

### Naty1

...yes

(This from Peter Bergmann, a student of Einsteins, THE RIDDLE OF GRAVITATION, PGS 60-61)

(He made several different formulations and only after he developed a new point of view via the Equivalence Principle was he able to select one he believed correct.)

5. Jan 2, 2009

### Heimdall

ok one more question : can you explain me a bit more this paragraph :

6. Jan 3, 2009

7. Jan 3, 2009

### dx

The reason SR can't handle gravity is because when gravity is present, it is not possible to find inertial reference frames for finite regions of spacetime. SR works only in inertial reference frames.

8. Jan 3, 2009

### Naty1

Heimdall....what is the source of your quote in post #5??

I'd like to read some before and after if I can find your portion....Right now I am having trouble with it...

9. Jan 3, 2009

### JesseM

Googling shows it's from Can Special Relativity handle accelerations? from the Physics FAQ hosted on John Baez's site. The equivalence principle analysis of the twin paradox from the twin paradox page on Baez's site may also be helpful in understanding how accelerating frames work in flat spacetime, and why they involve "pseudo gravitational fields".