# Relativistic Dynamics

So far we have said nothing about how to predict motion in relativity. Do Newton's laws still work? Do conservation laws still apply? The answer is yes, but many of the definitions need to be modified, and certain entirely new phenomena occur, such as the conversion of mass to energy and energy to mass, as described by the famous equation *E*=*mc*^{2}.

### Combination of velocities

The impossibility of motion faster than light is a radical difference between relativistic and nonrelativistic physics, and we can get at most of the issues in this section by considering the flaws in various plans for going faster than light. The simplest argument of this kind is as follows. Suppose Janet takes a trip in a spaceship, and accelerates until she is moving at 0.8*c* (80% of the speed of light) relative to the earth. She then launches a space probe in the forward direction at a speed relative to her ship of 0.4*c*. Isn't the probe then moving at a velocity of 1.2 times the speed of light relative to the earth?

The problem with this line of reasoning is that although Janet says the probe is moving at 0.4*c* relative to her, earthbound observers disagree with her perception of time and space. Velocities therefore don't add the same way they do in Galilean relativity. Suppose we express all velocities as fractions of the speed of light. The Galilean addition of velocities can be summarized in this addition table:

The derivation of the correct relativistic result requires some tedious algebra, which you can find in my book *Simple Nature* if you're curious. I'll just state the numerical results here:

Janet's probe, for example, is moving not at 1.2*c* but at 0.91*c*, which is a drastically different result. The difference between the two tables is most evident around the edges, where all the results are equal to the speed of light. This is required by the principle of relativity. For example, if Janet sends out a beam of light instead of a probe, both she and the earthbound observers must agree that it moves at 1.00 times the speed of light, not 0.8+1=1.8. On the other hand, the correspondence principle requires that the relativistic result should correspond to ordinary addition for low enough velocities, and you can see that the tables are nearly identical in the center.

### Momentum

Here's another flawed scheme for traveling faster than the speed of light. The basic idea can be demonstrated by dropping a ping-pong ball and a baseball stacked on top of each other like a snowman. They separate slightly in mid-air, and the baseball therefore has time to hit the floor and rebound before it collides with the ping-pong ball, which is still on the way down. The result is a surprise if you haven't seen it before: the ping-pong ball flies off at high speed and hits the ceiling! A similar fact is known to people who investigate the scenes of accidents involving pedestrians. If a car moving at 90 kilometers per hour hits a pedestrian, the pedestrian flies off at nearly double that speed, 180 kilometers per hour. Now suppose the car was moving at 90 percent of the speed of light. Would the pedestrian fly off at 180% of *c*?

To see why not, we have to back up a little and think about where this speed-doubling result comes from. For any collision, there is a special frame of reference, the center-of-mass frame, in which the two colliding objects approach each other, collide, and rebound with their velocities reversed. In the center-of-mass frame, the total momentum of the objects is zero both before and after the collision.

Figure q/1 shows such a frame of reference for objects of very unequal mass. Before the collision, the large ball is moving relatively slowly toward the top of the page, but because of its greater mass, its momentum cancels the momentum of the smaller ball, which is moving rapidly in the opposite direction. The total momentum is zero. After the collision, the two balls just reverse their directions of motion. We know that this is the right result for the outcome of the collision because it conserves both momentum and kinetic energy, and everything not forbidden is mandatory, i.e., in any experiment, there is only one possible outcome, which is the one that obeys all the conservation laws.

*self-check:*

How do we know that momentum and kinetic energy are conserved in figure q/1?

Let's make up some numbers as an example. Say the small ball has a mass of 1 kg, the big one 8 kg. In frame 1, let's make the velocities as follows:

| | |

| before the collision | after the collision |

| | |

small ball | -0.8 | 0.8 |

big ball | 0.1 | -0.1 |

| | |

Figure q/2 shows the same collision in a frame of reference where the small ball was initially at rest. To find all the velocities in this frame, we just add 0.8 to all the ones in the previous table.

| | |

| before the collision | after the collision |

| | |

small ball | 0 | 1.6 |

big ball | 0.9 | 0.7 |

| | |

In this frame, as expected, the small ball flies off with a velocity, 1.6, that is almost twice the initial velocity of the big ball, 0.9.

If all those velocities were in meters per second, then that's exactly what happened. But what if all these velocities were in units of the speed of light? Now it's no longer a good approximation just to add velocities. We need to combine them according to the relativistic rules. For instance, the table on page 87 tells us that combining a velocity of 0.8 times the speed of light with another velocity of 0.8 results in 0.98, not 1.6. The results are very different:

| | |

| before the collision | after the collision |

| | |

small ball | 0 | 0.98 |

big ball | 0.83 | 0.76 |

| | |

We can interpret this as follows. Figure q/1 is one in which the big ball is moving fairly slowly. This is very nearly the way the scene would be seen by an ant standing on the big ball. According to an observer in frame r, however, both balls are moving at nearly the speed of light after the collision. Because of this, the balls appear foreshortened, but the distance between the two balls is also shortened. To this observer, it seems that the small ball isn't pulling away from the big ball very fast.

Now here's what's interesting about all this. The outcome shown in figure q/2 was supposed to be the only one possible, the only one that satisfied both conservation of energy and conservation of momentum. So how can the *different* result shown in figure r be possible?

The answer is that relativistically, momentum must not equal *mv*. The old, familiar definition is only an approximation that's valid at low speeds. If we observe the behavior of the small ball in figure r, it looks as though it somehow had some extra inertia. It's as though a football player tried to knock another player down without realizing that the other guy had a three-hundred-pound bag full of lead shot hidden under his uniform --- he just doesn't seem to react to the collision as much as he should. This extra inertia is described by redefining momentum as

*m*γ

*v*.

At very low velocities, γ is close to 1, and the result is very nearly *mv*, as demanded by the correspondence principle. But at very high velocities, γ gets very big --- the small ball in figure r has a γ of 5.0, and therefore has five times more inertia than we would expect nonrelativistically.

This also explains the answer to another paradox often posed by beginners at relativity. Suppose you keep on applying a steady force to an object that's already moving at 0.9999*c*. Why doesn't it just keep on speeding up past *c*? The answer is that force is the rate of change of momentum. At 0.9999*c*, an object already has a γ of 71, and therefore has already sucked up 71 times the momentum you'd expect at that speed. As its velocity gets closer and closer to *c*, its γ approaches infinity. To move at *c*, it would need an infinite momentum, which could only be caused by an infinite force.

### Equivalence of mass and energy

Now we're ready to see why mass and energy must be equivalent as claimed in the famous *E*=*mc*^{2}. So far we've only considered collisions in which none of the kinetic energy is converted into any other form of energy, such as heat or sound. Let's consider what happens if a blob of putty moving at velocity *v* hits another blob that is initially at rest, sticking to it. The nonrelativistic result is that to obey conservation of momentum the two blobs must fly off together at *v*/2. Half of the initial kinetic energy has been converted to heat.^{7}

Relativistically, however, an interesting thing happens. A hot object has more momentum than a cold object! This is because the relativistically correct expression for momentum is *m*γ *v*, and the more rapidly moving atoms in the hot object have higher values of γ. In our collision, the final combined blob must therefore be moving a little more slowly than the expected *v*/2, since otherwise the final momentum would have been a little greater than the initial momentum. To an observer who believes in conservation of momentum and knows only about the overall motion of the objects and not about their heat content, the low velocity after the collision would seem to be the result of a magical change in the mass, as if the mass of two combined, hot blobs of putty was more than the sum of their individual masses.

Now we know that the masses of all the atoms in the blobs must be the same as they always were. The change is due to the change in γ with heating, not to a change in mass. The heat energy, however, seems to be acting as if it was equivalent to some extra mass.

But this whole argument was based on the fact that heat is a form of kinetic energy at the atomic level. Would *E*=*mc*^{2} apply to other forms of energy as well? Suppose a rocket ship contains some electrical energy stored in a battery. If we believed that *E*=*mc*^{2} applied to forms of kinetic energy but not to electrical energy, then we would have to believe that the pilot of the rocket could slow the ship down by using the battery to run a heater! This would not only be strange, but it would violate the principle of relativity, because the result of the experiment would be different depending on whether the ship was at rest or not. The only logical conclusion is that all forms of energy are equivalent to mass. Running the heater then has no effect on the motion of the ship, because the total energy in the ship was unchanged; one form of energy (electrical) was simply converted to another (heat).

The equation *E*=*mc*^{2} tells us how much energy is equivalent to how much mass: the conversion factor is the square of the speed of light, *c*. Since *c* a big number, you get a really really big number when you multiply it by itself to get *c*^{2}. This means that even a small amount of mass is equivalent to a very large amount of energy.

### Contributors

- Benjamin Crowell,
**Conceptual Physics**