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 specifically in special relativity? (I mean that I have not studied general relativity yet) What stay constant ? Or are we making an approximation?

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

I hope my question has a 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

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 3

No one mentioned The Relativistic Rocket? Those are the special relativistic equations for acceleration.

Answer 4

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