Adding Velocities the Relativistic Way

The Intuition Behind Relativistic Addition

Imagine two observers, Alice and Bob. Alice is stationary on Earth, and Bob is in a spaceship moving at a significant fraction of the speed of light relative to Alice. Bob throws a ball forward. In Bob's frame, the ball moves at speed \(u'\). In Alice's frame, the ball moves at speed \(u\). If we used simple addition, \(u = v + u'\), we might predict the ball exceeds \(c\). Relativity prevents this by "dilating" the time and length measurements between the frames. The denominator in the relativistic formula acts as a correction factor that becomes significant only when \(v\) or \(u'\) are close to \(c\). At low speeds, this correction is negligible, and we recover the familiar Galilean result. This ensures that the speed of light remains the universal speed limit, acting as an asymptote that massive particles can approach but never cross.

Deriving the Longitudinal Component

To find the velocity in frame \(S\), we start with the definition of velocity \(u_x = \frac{dx}{dt}\). Using the Lorentz transformations for space and time, we express \(dx\) and \(dt\) in terms of the primed coordinates. The spatial transformation is \(dx = \gamma(dx' + v dt')\) and the temporal transformation is \(dt = \gamma(dt' + \frac{v}{c^2} dx')\). Dividing these expressions gives the ratio \(\frac{dx}{dt}\).

Dividing the numerator and denominator by \(dt'\) allows us to substitute \(u_x' = \frac{dx'}{dt'}\). The \(\gamma\) factors cancel out, leaving the standard relativistic velocity addition formula for the parallel component.

This equation shows that if \(u_x' = c\), then \(u_x = \frac{c+v}{1+v/c} = c\). The speed of light is invariant. If \(u_x' < c\) and \(v < c\), the result is always less than \(c\).

Transverse Velocity Components

Velocities perpendicular to the direction of relative motion also change, but in a different way. For the \(y\)-component, we use \(u_y = \frac{dy}{dt}\). Since the Lorentz transformation does not mix \(y\) and \(t\) directly in the same way, we have \(dy = dy'\). However, \(dt\) still contains the \(dx'\) term. Substituting \(dt = \gamma(dt' + \frac{v}{c^2} dx')\) into the velocity definition yields a more complex expression.

Recognizing that \(\frac{dy'}{dt'} = u_y'\) and \(\frac{dx'}{dt'} = u_x'\), we arrive at the transverse velocity transformation. Note the presence of \(\gamma\) in the denominator and the coupling with the longitudinal velocity \(u_x'\).

The same logic applies to the \(z\)-component. This result implies that even if an object moves purely in the \(y\)-direction in its own frame (\(u_x' = 0\)), an observer in \(S\) will see a reduced \(y\)-velocity due to time dilation (\(u_y = u_y'/\gamma\)).

Consistency with the Speed of Light

A crucial test of any relativistic formula is its behavior for light. Consider a photon moving in the \(x\)-direction in frame \(S'\), so \(u_x' = c\) and \(u_y' = u_z' = 0\). Applying the longitudinal formula:

The result is exactly \(c\), independent of \(v\). This confirms that the speed of light is invariant under Lorentz transformations. For massive particles, where \(u' < c\), the denominator \(1 + \frac{u' v}{c^2}\) is always greater than 1 (assuming \(v\) and \(u'\) are in the same direction), ensuring that \(u\) remains less than \(c\). This mathematical structure enforces the causal structure of spacetime, preventing information or matter from traveling faster than light.