Acceleration in special relativity

Question

I am currently studying the motion of relativistic charged particles in electromagnetic fields. More exactly, we first derived the equation of motion in the 4-vector formalism.

I was a bit confused when my teacher talked about acceleration in special relativity. In which cases are we allowed to talk about it in special relativity? (I have not studied general relativity yet) What stays constant ? Are we making an approximation?

It may, of course, make sense, though I have the feeling that, in my courses, we only talked about constant speeds in special relativity.

I hope my question makes sense.

Isaac

Answer 1

Acceleration is an acceptable topic in special relativity. It is rather easily described. The metric for flat spacetime is $$ ds^2~=~-dt^2~+~dx^2~+~dy^2~+~dz^2~=~g_{ab}dx^adx^b. $$ If I divide through by the square of the proper time $ds^2$ this gives unity $$ 1~=~g_{ab}\frac{dx^a}{ds}\frac{dx^b}{ds}~=~g_{ab}U^aU^b. $$ A derivative with respect to the proper time $s$ gives 0 on the left hand side and the spacetime acceleration $A^a$ is clearly seen to be orthogonal to the four velocity $U^a$.

If I restrict this to two dimensions, where the spatial dimension of importance is the direction the object moves along the metric in a change of signature convention gives us $$ 1~=~(U^t)^2~-~(U^x)^2 $$ The equation implies that the two components of the four-velocity are hyperbolic trigonometric functions $$ U^x~=~\sinh gs,~U^t~=~\cosh gs, $$ for $g$ the acceleration parameter. The motion of this body asymptotes to a null direction $u~=~t~-~x$ and is a hyperboloid restricted to this one part of the Minkowski spacetime. This portion is called the Rindler wedge, which has all sorts of surprising structure to it, from Fermat coordinate representation that is a form of Poincare half plane, to Unruh radiation.

There is some confusion with respect to accelerated frames in special relativity. It is commonly thought that it is the domain of general relativity. Special relativity should be thought of as Newton’s laws with the extension of boost symmetries in space plus time. Then just as with Newton’s laws the symmetries are properly described from an inertial frame, and the dynamics are $F~=~ma$. The description of accelerated motion for Newton’s second law in special relativity is perfectly acceptable physics.

Answer 2

There are three things you might want to do using relativity: (1) describe an object that's accelerating in flat spacetime; (2) adopt a frame of reference, in flat spacetime, that's accelerating; (3) describe curved spacetime. General relativity is only needed for #3.

A prohibition on #1 is particularly silly. It would make SR into a trivial theory incapable of describing interactions. If you believed this, you would have to stop believing, for example, in the special-relativistic description of the Compton effect and fine structure in hydrogen; these phenomena would have to be described by some as yet undiscovered theory of quantum gravity.

Number 1 often comes up in discussions of the twin paradox. A good way to see that general relativity is totally unnecessary for understanding the twin paradox is to pose a version in which the four-vector equation a=b+c represents the unaccelerated twin's world-line a and the accelerated twin's world-line consisting of displacements b and c. The accelerated twin is subjected to (theoretically) infinite accelerations at the vertices of the triangle. The triangle inequality for flat spacetime is reversed compared to the one in flat Euclidean space, so proper time |a| is greater than proper time |b|+|c|.

Number 2, accelerated frames, is less trivial. It's for historical reasons that you'll see statements that SR can't handle accelerated frames. Einstein published special relativity in 1905, general relativity in 1915. During that ten-year period in between, nobody really knew what the boundaries of applicability of special relativity were. This uncertainty made its way into textbooks and lectures, and because of the conservative nature of education, some students are still hearing, a century later, incorrect assertions about it. There is an overwhelming consensus among modern relativists that the boundary between SR and GR should be defined as the distinction between flat and curved spacetime, not unaccelerated and accelerated observers.[MTW 1973,Penrose 2004,Taylor 1992,Schutz 2009,Hobson 2005]

In an accelerating frame, the equivalence principle tells us that measurements will come out the same as if there were a gravitational field. But if the spacetime is flat, describing it in an accelerating frame doesn't make it curved. (Curvature is invariant under any smooth coordinate transformation.) Thus relativity allows us to have gravitational fields in flat space --- but only for certain special configurations like uniform fields. SR is capable of operating just fine in this context. For example, Chung et al. did a high-precision test of SR in 2009 using a matter interferometer in a vertical plane, specifically in order to test whether there was any violation of Lorentz invariance in a uniform gravitational field. Their experiment is interpreted purely as a test of SR, not GR.

MTW 1973 -- Misner, Thorne, and Wheeler, Gravitation, 1973, p. 163: "Accelerated motion and accelerated observers can be analyzed using special relativity." p. 164: "An accelerated observer can carry clocks and measuring rods with him, and he can use them to set up a reference frame (coordinate system) in his neighborhood."

Penrose, The Road to Reality, 2004, p. 422, "It used to be frequently argued that it would be necessary to pass to Einstein's general relativity in order to handle acceleration, but this is completely wrong. [...] We are working in special relativity provided that [the] metric is the flat metric of Minkowski Geometry M."

Taylor and Wheeler, Spacetime Physics, 1992, p. 132: "DO WE NEED GENERAL RELATIVITY? NO! [...] 'Don't you need general relativity to analyze events in accelerated reference frames?' 'Oh yes, general relativity can describe events in the accelerated frame,' we reply, 'but so can special relativity if we take it in easy steps!'"

Schutz, A First Course in General Relativity, 2009. Schutz equivocates on pp. 3 and 141 about the status of accelerated observers in SR, but says, "[...] the real physical distinction between these two theories is that special relativity (SR) is capable of describing physics only in the absence of gravitational fields, while general relativity (GR) extends SR to describe gravitation itself."

Hobson, General Relativity: An Introduction for Physicists, 2005, sec. 1.14, discusses "Event horizons in special relativity" from the point of view of accelerated observers, using coordinates defined in their accelerated reference frames.

Chung -- http://arxiv.org/abs/0905.1929

Answer 3

It's perfectly possible to have accelerated particles in special relativity in the same way you can have them in classical Newtonian physics.

However, to describe them one needs some of the tools and concepts of differential geometry used commonly in general relativity. For example, if you'd like to describe physics in the reference frame of the accelerating observer, this is no longer inertial and you can't use the simple math of linear Lorentz transformations anymore because you also need to take into account non-linear effects (some of which are present already in classical case, like centrifugal and Coriolis forces).

So what you should do is describe the world-line of the particle as a curve $\gamma(\lambda)$ (parametrized by some affine parameter). You can then compute the velocity of the particle ${\mathbf v} = {{\rm \mathbf d}\gamma(\lambda) \over {\rm d} \lambda}$ as a tangent field along the world-line and the acceleration ${\mathbf a} = {{\rm D} {\mathbf v} \over {\rm d} \lambda} $ as a covariant derivative of velocity along the world-line; here we are implicitly using that we have a flat connection available on our manifold. Working backwards, you can postulate that the particle is to have some given acceleration due to forces and figure out its trajectory (same way you do it in classical physics).

If you'd like to figure out additional information you can choose some coordinate system at point $\lambda = \lambda_0$. Then you can use Fermi-Walker transport to transport this system along the world-line which gives you information about the way the particle is spinning, etc.

Answer 4

To Isaac's question - yes, your question has a HUGE amount of sense in it. It is the fundamental question of relativity - can SR describe a situation with something undergoing acceleration?

Everyone always makes the statement that the answer is yes as long as we use an inertial frame. But then they go and proceed with the following: the invariant world-path (since one frame is proper) $$ds^2 = d\tau^2 = dt^2 - (dx^2 + dy^2 + dz^2)$$ and from this they proceed to derive the twin paradox by dividing through by $dt^2$ and then integrating to get an equation that shows time dilation (but perhaps without separating out the dilation due to v/c effects from the real dilation due to the acceleration. But what concerns me is that I don't think $ds^2$ is invariant since it's based on the constancy of the speed of light (it's basically a light-beam as seen from two inertial observers). Isn't this "cheating"? Or is that most people are doing this at the differential level along a path and then doing what should be a very careful integral along that path (even if now converted to definite integrals over time). One frame is purely moving in time (the proper frame) so if $c$ is taken as 1, $ds$ is $d\tau$ (i.e. a differential of proper time). This makes it easy to set up a comparison between the times elapsed. But I think it's wrong to do this unless we talk very carefully about the invariance of worldlines. what I would like is someone or me to prove somehow that we can take a limit to an infinitesimal and somehow still use the expanding light beam (which is basically just the Lorentz transformation). I worry a lot about this since if something is accelerating, the speed of light is no longer a constant. Can someone pick one of the above references that really nails this idea.

Schutz is probably correct when he says that we need to use GR. Honestly, I think he's right but I'm not entirely sure. If I have to spend the rest of my life on this, I will, but Jesus, it's a lot of work to prove all this crap. We need to derive the integral of $(1 - v(t)^2)$ over time and prove that it's correct. I don't think it is but it appears all over the place.

If you can answer this, I'll tip my hat - I'm thinking I need to jump right into Einstein-Hilbert for some sort of magic field-consistency so light can change speed. I can't see SR solving these problems unless we stretch it like a membrane - I believe Lawrence B. Crowell was doing this - is his analysis SR?

Marek above makes a lot of sense. He knows that we can't just use an invariant $ds^2$ (as it is the Lorentz transformation). I like his answer.

Please forgive my two posts if it's obvious that we can use Lorentz invariance of $ds^2$ since it's infinitestimal. I just need to make sure - hence my interest in comparing to GR. I am not trying to analyze this from the non-inertial frame (at least yet).

http://users.telenet.be/vdmoortel/dirk/Physics/Acceleration.html If we look at the approach discussed at the above link, we are analyzing with the acceleration "felt" in the accelerating frame (this keeps things realistic). Notice how easy it was with four-vectors. I had originally simply planned to do it from the point of view of a constant acceleration relative to the INTERTIAL observer (which makes the problem very easy to set up). So we only need to grind through one integral. But it would probably illustrate the same concept of time dilation, just quantitatively different. So I will do it the easy way, now that I've read the more realistic (and more complex due to the use of the velocity addition formula in differential form) way to solve the problem. So his formalism is useful. I guess we are hinting at Euler-Lagrange here too so we are not far from the GR. So to get my feet wet, I will just grind through the simpler way - frankly, I don't need to repeat dirk van de moortel's other than to read it so I understand what he's doing. So thanks to Dirk for his analysis. Of course our analyses hammer home the point that most (if not all) of the time dilation effect (even with acceleration) enters through the velocity (or better, the speed) so this will help me understand and/or convince myself of that point too. His use of constant in rocket's frame is what a passenger would want - a non-changing force on his body, to better simulate earth.