Introduction to Electrodynamics · Problem 10.17

Potentials and Fields — David J. Griffiths

Physics · hyperbolic-motion-horizon

Problem

I showed that at most one point on the particle trajectory communicates with r\mathbf{r} at any given time. In some cases there may be no such point (an observer at r\mathbf{r} would not see the particle---in the colorful language of general relativity, it is ``over the horizon''). As an example, consider a particle in hyperbolic motion along the xx axis: w(t)=b2+(ct)2 x^(−∞<t<∞).(10.52)\mathbf{w}(t) = \sqrt{b^2 + (ct)^2}\,\hat{\mathbf{x}} \quad (-\infty < t < \infty). \tag{10.52} (In special relativity, this is the trajectory of a particle subject to a constant force F=mc2/bF = mc^2/b.) Sketch the graph of ww versus tt. At four or five representative points on the curve, draw the trajectory of a light signal emitted by the particle at that point---both in the plus xx direction and in the minus xx direction. What region on your graph corresponds to points and times (x,t)(x,t) from which the particle cannot be seen? At what time does someone at point xx first see the particle? (Prior to this the potential at xx is zero.) Is it possible for a particle, once seen, to disappear from view?
Intuitive explanation

This problem explores the causal structure of spacetime and how the finite speed of light creates "horizons" that limit what observers can see. The key insight is that light signals travel at ±c\pm c, so from any spacetime point (x,t)(x,t), you can only receive signals from the particle if its trajectory intersects the backward light cone—the pair of lines with slopes ±1/c\pm 1/c emanating from that point. By sketching the hyperbolic worldline and these light cones, you'll discover that the accelerating particle leaves an inaccessible region behind it, and that once an observer sees the particle, they continue to see it forever (causality prevents "disappearing").

Intuitive explanation

This problem explores the causal structure of spacetime and how the finite speed of light creates "horizons" that limit what observers can see. The key insight is that light signals travel at ±c\pm c, so from any spacetime point (x,t)(x,t), you can only receive signals from the particle if its trajectory intersects the backward light cone—the pair of lines with slopes ±1/c\pm 1/c emanating from that point. By sketching the hyperbolic worldline and these light cones, you'll discover that the accelerating particle leaves an inaccessible region behind it, and that once an observer sees the particle, they continue to see it forever (causality prevents "disappearing").

Related theory

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