Acceleration, Relativity and Distance

[Walt Sellers]

As I recall, since light bends in an accelerating frame of reference, a light beam will bend with the acceleration. Also, the same light beam would define the relativistic straight-line path between two points it crosses.

For the question, the proverbial space ship is sitting still with a laser pointer on one side lighting up a target point on the other side. When the ship accelerates, the beam bends down and is no longer lighting the target. (The amount of bend isn't important, just that it is bending.) Now, to light the target again, the laser pointer must be aimed "up" against the acceleration. The beam is now an arch to reach the same target point.


Question: Does this mean the begin and end points of the laser path are farther away from each other while accelerating than while sitting? (In Relativistic terms.)

[Ken G]

Very tricky question. One might need to be a real expert in relativity to answer it, but I'll give you my take. I would say that you have to specify quite clearly what length you want to get, but if you are talking about the spatial length of the path taken by a laser pulse in the accelerating frame, then yes, it increases due to the curved path. However, this length is just what is known as a "coordinate length", meaning it depends on how you are coordinatizing the situation, which here means it only comes out a meaningful number for people on the spaceship who choose spatial coordinates that are pegged to the ship.

If, on the other hand, you are seeking a kind of invariant length, one that holds in an absolute way that does not depend on the coordinates, it's a lot trickier. There is an event when the laser emitted the pulse, and an event where the pulse is absorbed at the other side, and there is a single concept of "separation" between those events that is invariant (so does not depend on reference frame). Unfortunately, the invariant separation at the ends of a light path is always zero-- the path is a "null geodesic."

OK, so the invariant separation between the events of emission and absorption is zero, but that's not quite what you asked, you wanted the distance between the two points, and each of those points has its own "world line", or path through spacetime. So perhaps when one is asking for a distance between two points, one is seeking an invariant separation between world lines, not between events. This brings us to the concept of proper distance, which is the invariant separation between two events that are deemed to be simultaneous in some frame, and are given by the spatial distance between the events as reckoned in the frame where they are simultaneous. So we might imagine picking the frame of one of the points, say the laser pointer, and asking for the distance to the absorption point when the absorption point is deemed simultaneous with the emission (which of course comes at a time before the emission is received).

Now, I'm not sure if this can be done in general from the accelerating frame of the laser pointer, but we can certainly choose an instantaneously comoving inertial frame, and in that frame, there will be a point on the world line of the absorption point that is deemed to be simultaneous with the emission event (we might imagine a pulse of laser light, for example, and we are concerned with when it is emitted but not with when it is received). In the comoving inertial frame, the event deemed simultaneous with the emission is the place on the receiver's world line that is "straight across from" the laser pointer. Thus the "proper distance" between those points is still just the width of the ship, there's no change due to the acceleration. So if you are asking for the "proper distance between laser and receiver", you can indeed get an invariant length, and it will not be different due to the acceleration of the ship. However, it has nothing to do with the spacetime path of the laser pulse itself, as that is still a "null geodesic", and the only invariant separation it gives rise to comes out zero. Also, it has nothing to do with the actual "path of a laser pulse", because the simultaneous event we are using to get a proper distance is not the event of receipt of the laswer pulse, so it is not at "the other end of the bridge", as it were. So I don't think that's actually what you are asking about-- I think you are asking about the type of coordinate distance I was talking about originally. But the bottom line is, things get complicated, and precise language is both difficult and crucial.

Welcome to the forum, where you will often find that the questions are better than they seem, and the answers are worse!