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There are two theories of relativity; special and general. The special theory only deals with one kind of motion: the motion at constant velocity. The general theory, which Einstein took another 10 years to develop, deals with all motion in spacetime. It turns out to be a theory about gravity, and, as such, it has a very different "flavor," which we will discuss very briefly at the end of the class.

What Einstein did with special relativity is to propose to extend the old relativity principle to all laws of physics (not just mechanics).

Principle of Relativity (ala Newton): All the laws of physics (not just mechanics) are the same for all observers moving at constant velocity.

From this principle, the theory of special relativity follows. Here is the logic for special relativity:

1. Maxwell's theory of electricity is true for all observers moving at constant speed.
2. Maxwell's theory predicts that light travels at c= 299,792,458\ m/s .
3. Therefore, light travels at speed c as seen by all observers moving at constant speed .

• M: This means that for any experiments, irrespective of the speed the observers are moving, the observer will measure the speed of light to be c.
• S: But this is so weird... Let me try this setup. Let's say a train is moving at 40 mph with respect to the ground and then another one is coming head on at also 40 mph with respect to ground. From the reference frame of the first moving train don't you see the other train coming at you with 80 mph? -M: Yes, but here's Einstein's idea: if you are on one train, and you shine a flashlight towards the other train, the speed of the light relative to the ground is still c (not c+ 40 mph).
• S: Wierd...
• M: OK back to your example of the 2 trains approaching each other each traveling 40 mph with respect to ground. This is the addition of velocity that we saw earlier. It works pretty well at low speed (like 40 mph), but, if you move close to the speed of light, it fails. Looks at the figure below. Let us say that the first train was going at 0.8 c (0.8 times the speed of light) and the other one was in collision course also at 0.8 c. By Newton's logic, the velocity of train 2 with respect to train 1 would be 1.6 c. But it cannot be, because nothing moves faster than light!

How to add speeds: ** this is optional **, your book also discusses this.

Einstein's principle of relativity says that the speed of light is measured to be the same for all observers moving at constant velocity. Lets go back to Marie's example: Imagine I turn on a flashlight in a train moving at the speed of c/2 with respect to the ground v_{TG} = c/2 , the speed of the light with respect to the train v_{cT} is c and the speed of light with respect to the ground v_{cG} must also be c.

Clearly the speed of light does not satisfy the simple addition of speed that we have seen before.

v_{cG} \neq v_{cT}+v_{TG}

We need a new formula. It must be a formula that keeps the speed constant. It's a fun little math problem to find a way to add velocity such that the speed of light always stays the same. Here is a way (the correct one) to do it (** You do not need to remember or be able to use this formula ):

v_{cG} = \frac{v_{cT}+v_{TG}}{1+\frac{v_{cT}v_{TG}}{c^2}}

It's almost just the same as before but we now divide by the factor 1+\frac{v_{cT}v_{TG}}{c^2} . So if the speed of the light with respect to the train is c and the speed of the train is c/2 we find that the speed of light with respect to ground is

v_{cG} = \frac{c+c/2}{1+\frac{c^2}{2 c^2}} = c

In fact for any speed of the train the speed of light is always c in any reference frame. Even if the train moves at the speed of light, the speed of light inside the train remains c with respect to the ground or with respect to the train. Also this formula shows that if the two velocity you are adding are less than c, than the resulting "sum" is always less than c.

If you move slower than the speed of light, you will always move slower than the speed of light. In any reference frame. Since we have never seen anything move faster than light. It is natural to assume that this is a speed limit.