FIGURE 36.34a is a graph of momentum versus velocity. For a Newtonian particle, with $p=mu$, the momentum is directly proportional to the velocity. The relativistic expression for momentum agrees with the Newtonian value if $u\ll c$, but p approaches $\text{∞}$ as $u\to c$.

FIGURE 36.34 The speed of a particle cannot reach the speed of light.

The implications of this graph become clear when we relate momentum to force. Consider a particle subjected to a constant force, such as a rocket that never runs out of fuel. If F is constant, we can see from $F=dp/dt$ that the momentum is $p=Ft$. If Newtonian physics were correct, a particle would go faster and faster as its velocity $u=p/m=(F/m)t$ increased without limit. But the relativistic result, shown in FIGURE 36.34b, is that the particle’s velocity asymptotically approaches the speed of light $(u\to c)$ as p approaches $\text{∞}$. Relativity gives a very different outcome than Newtonian mechanics.

The speed c is a “cosmic speed limit” for material particles. A force cannot accelerate a particle to a speed higher than c because the particle’s momentum becomes infinitely large as the speed approaches c. The amount of effort required for each additional increment of velocity becomes larger and larger until no amount of effort can raise the velocity any higher.

Actually, at a more fundamental level, c is a speed limit for any kind of causal influence. If I throw a rock and break a window, my throw is the cause of the breaking window and the rock is the causal influence. If I shoot a laser beam at a light detector that is wired to a firecracker, the light wave is the causal influence that leads to the explosion. A causal influence can be any kind of particle, wave, or information that travels from A to B and allows A to be the cause of B.

For two unrelated events—a firecracker explodes in Tokyo and a balloon bursts in Paris—the relativity of simultaneity tells us that they may be simultaneous in one reference frame but not in others. Or in one reference frame the firecracker may explode before the balloon bursts but in some other reference frame the balloon may burst first. These possibilities violate our commonsense view of time, but they’re not in conflict with the principle of relativity.

For two causally related events—A causes B—it would be nonsense for an experimenter in any reference frame to find that B occurs before A. No experimenter in any reference frame, no matter how it is moving, will find that you are born before your mother is born. If A causes B, then it must be the case that $t_{\text{A}}<t_{\text{B}}$ in all reference frames.

Suppose there exists some kind of causal influence that can travel at speed $u>c$. FIGURE 36.35 shows a reference frame S in which event A occurs at position $x_{\text{A}}=0$. The faster-than-light causal influence—perhaps some yet-to-be-discovered “z ray”—leaves A at $t_{\text{A}}=0$ and travels to the point at which it will cause event B. It arrives at $x_{\text{B}}$ at time $t_{\text{B}}=x_{\text{B}}/u$.

FIGURE 36.35 Assume that a causal influence can travel from A to B at a speed $u>c$.

How do events A and B appear in a reference frame $\text{S}\prime$ that travels at an ordinary speed $v<c$ relative to frame S? We can use the Lorentz transformations to find out. Because $x_{\text{A}}=0$ and $t_{\text{A}}=0$, it’s easy to see that $x^{\prime }{}_{\text{A}}=0$ and $t^{\prime }{}_{\text{A}}=0$. That is, the origins of S and $\text{S}\prime$ overlap at the instant the causal influence leaves event A. More interesting is the time at which this influence reaches B in frame $\text{S}\prime$. The Lorentz time transformation for event B is

where we first factored out $t_{\text{B}}$, then made use of the fact that $u=x_{\text{B}}/t_{\text{B}}$ in frame S.

We’re assuming that $u>c$, so there exist ordinary reference frames, with $v<c$, for which $vu/c^2>1$. In that case, the term ($1-vu/c^2$) is negative and $t_{\text{B}}^{\prime }<0.$ But if $t_{\text{B}}^{\prime }<0.$ then event B happens before event A in reference frame $\text{S}\prime$. In other words, if a causal influence can travel faster than c, then there exist reference frames in which the effect happens before the cause. We know this can’t happen, so our assumption $u>c$ must be wrong. No causal influence of any kind—particle, wave, or yet-to-be-discovered z rays—can travel faster than c.

The existence of a cosmic speed limit is one of the most interesting consequences of the theory of relativity. “Warp drive,” in which a spaceship suddenly leaps to faster-than-light velocities, is simply incompatible with the theory of relativity. Rapid travel to the stars will remain in the realm of science fiction unless future scientific discoveries find flaws in Einstein’s theory and open the doors to yet-undreamed-of theories. While we can’t say with certainty that a scientific theory will never be overturned, there is currently not even a hint of evidence that disagrees with the special theory of relativity.