HPS 0410 | Einstein for Everyone |

Back
to Chapter, E=mc^{2}

John
D. Norton

Department of History and Philosophy of Science

We have seen in the main chapter that, according to classical physics, it is quite possible to use collisions to boost objects past the speed of light. Here we will review in a little more detail how this comes about. Then we will see how the mechanisms are blocked by relativity theory.

It will be helpful to look at one mechanism for applying a force to a body. A tiny mass floating freely in space is approached by a massive object moving at 100,000 miles per second. They collide elastically. The massive object's motion is barely affected. The tiny mass is boosted to 200,000 miles per second--greater than the speed of light. Here's how the collision looks to an observer on a nearby planet.

How do we know that the tiny mass is boosted to 200,000 miles per second by this collision? A very simple argument tells us this. First we imagine the collision as viewed by an observer standing on the huge mass:

Here's how things proceed from that observer's vantage point. The
tiny mass approaches at 100,000 miles per second and bounces off the
huge mass elastically. By symmetry it bounces
off at 100,000 miles per second with respect to the huge
mass.
The plus and the minus signs tell us the direction of the motion. "Plus" is to the right; "minus" is to the left. (Of course we are assuming the ideal case of a perfectly elastic collision in which the small mass loses none of its speed in the bounce.) |

We now transform back from the view of the observer on the huge mass to our the view of an observer on a nearby planet. The planet observer judges the system of two masses as a whole to have a velocity of +100,000, miles per second. So, to transform back to this observer's view, we merely add this 100,000 mile per second velocity to each of the velocities in the two mass system. Taking each component in turn:

The huge mass ends up with

0 + 100,000 = 100,00 miles per second.

The little mass prior to the collision ends up with

-100,000 + 100,000 = 0 miles per second.

The little mass after the collision ends up with

100,000 + 1000,000 = 200,000 miles per second.

This last addition is the important one, so let us look at it more carefully:

Final velocity of tiny mass with respect to planet 200,000 |
= | Final velocity of tiny mass with respect to big mass 100,000 |
+ | Velocity of big mass with respect to planet 100,000 |

Here's everything in a picture.

The above scheme for producing a velocity faster than light seems so simple. However, it must fail in relativity theory. How does it fail? By carefully repeating the above demonstration, we can see where the failure happens.

As before, imagine the collision from the point of view of an observer on the huge mass. The tiny mass approaches at 100,000 miles per second and bounces off at 100,000 miles per second. |

We now transform back to our original point of view on a nearby planet. To recover the final velocity of each component mass, we proceed as before. The planet observer judges the totality of the system of two masses as moving at 100,000 miles per second to the right. So we add this 100,000 miles per second to each velocity in the system. However--and here is the key point--we cannot use ordinary arithmetic addition to add this velocity. We must use the relativistic rule for composition of velocities that we saw earlier.

When we do that everything works out as before for the velocity of the tiny mass before the collision and the huge mass. But it does not work out the same for the velocity of the tiny mass after the collision. To find that velocity we must compose the 100,000 miles per second of the tiny mass with respect to the huge mass with the 100,000 miles per second of the huge mass with respect to the planet. When they are composed by the relativistic rule, we do not end up with a result greater than the speed of light. They compose to 155,000 miles per second.

Final velocity of tiny mass with respect to planet 155,000 mi/sec |
= | Final velocity of tiny mass with respect to big mass 100,000 mi/sec |
adding by relativistic composition |
Velocity of big mass with respect to planet 100,000 mi/sec |

Here's a picture that shows the sums.

The final outcomes of the compositions are:

It is important to see what this last analysis does and does not show.

It does show that the kinematics of special relativity blocks this particular mechanism from accelerating bodies to speeds greater than that of light.

It does not show how our ideas about energy, momentum and forces must be adjusted so that they do not allow for the acceleration of bodies through the speed of light. E=mc^{2} arises from those adjustments.

So far we have analysed the simplest case of the collision
between a large moving mass and a small mass at rest. In the chapter, we
considered the slightly more complicated case
of a small mass already moving that then collides with a large mass. That
was the case of the two massive bodies on rails approaching a small mass
that bounced back and forth between them.

The collisions of this example can be analysed by the same technique. Take
each collision one at a time. Transform to the frame
of reference of the block involved. In it, the mass will approach
the block, which is at rest, and bounce off the block with the same speed,
but in the opposite direction. Now transform back to the original frame of
reference to recover the outcome of the collision. If the relativistic
rule of composition of velocities is used in these transformations, no
bounce will succeed in boosting the little mass past the speed of light.
For every composition will be of velocities less than that of light. They
can never be composed by the relativistic rule to produce a velocity
greater than light's.

Copyright John D. Norton.January 2001, January 11, September 23, 2008. January 25, 2022. January 28, 2024