Uniform Acceleration

Suppose an object starts at rest with respect to the lab frame and undergoes a uniform acceleration a′ as measured by an observer on a spaceship moving at a uniform velocity v with respect to the lab. In Newtonian mechanics, for speeds where v << c, the velocity after t ′ seconds in the moving frame has elapsed is V′ = a′ t ′ as measured by the observer on the moving object. This velocity is V = v + a′ t after the elapsed time t in the lab frame, because in Newtonian physics the clocks in the different frames run at the same rates. What is the velocity value in the lab frame when the speed is allowed to become relativistic? Can the product at be greater than c in either reference frame?

Answer

In the STR, the velocity in the lab frame is no longer V = a′t for a uniform acceleration a′ in the moving frame. However, in the moving frame at each instant the expression V′ = a′t ′ continues to be true. To convert from the moving frame to the lab frame, we must essentially convert the clock readings and time interval using dt/dτ . Here, τ is the proper time that is, the clock reading on a wristwatch worn by an observer on the spaceship, say, and dτ is the proper time interval between two events at the same location. In the example, τ is the elapsed time on the wristwatch of the person on the moving frame. Hence, on the moving spaceship frame, V′ = a′τ.

Before we determine the answer for the velocity of the object in the lab frame, let’s review the simpler problem of how velocities are added in relativistic frames. If an object moves forward with the velocity V′ in the spaceship frame, then the object’s velocity V in the lab frame is determined by the law of addition of velocities V/c = (V′/c + V_s/c)/(1 + V′V_s/c²), where Vs is the uniform velocity of the spaceship in the lab frame. One can check the limiting case for low velocities, when V′V_s/c² is very small, to verify agreement with Galilean relativity that is, the two velocities simply add. To relate the acceleration of the object as seen by both observers, the addition of velocities expression is differentiated with respect to the time in the lab reference frame to obtain a = a′ /{(1 + V′V_s/c²) }, a messy expression. The velocity of the accelerating object in the lab frame is found by substituting V′ = a′ τ. Therefore a ≠ a′ and V < c.

An alternative mathematical technique using a velocity parameter defined in terms of hyperbolic functions is given in the Taylor and Wheeler reference below.