Classical Forward Derivation

Classical Travel of Light from A to B

Here we derive the time interval t_{_B} - t_{_A} for the travel of the photon from A to B for a source of light at rest with the stationary system.

During the movement of the rod its end A moves according to the following classical law of motion

A(t) = A(t_{_A}) + v(t - t_{_A}),

the point A being at point A(t_{_A}) in the stationary system at the moment t_{_A}.

Since we are observing light (photon) emitted from a stationary source in the stationary system, then, according to the result from the experiment of the Michelson-Morley experiment, its velocity is the same in all directions of the stationary system and is equal to c.

This has to be specially underlined - the velocity of the photon in the stationary system M is c = const due to purely classical reasons having nothing to do with the STR.

The impression of some, mislead by the implications in §1 of 1905 paper, that velocity of light c = const implies application the second postulate is erroneous. The second postulate of STR has no role whatsoever here.

Then, purely classically (without STR), the law of motion of the photon before reaching B is

F(t) = F(t_{_A}) + c(t - t_{_A})

where the photon F is in point F(t_{_B}) at the moment t_{_B}

We know, from the condition of the problem, that classically, the photon F at the moment tB will be in the point

F(t_{_B}) = F(t_{_A}) + c(t_{_B} - t_{_A})

(recall that for any moment t the photon F will be in the point F(t) = F(t_{_A}) + c(t - t_{_A}) or
F(t) = A(t_{_A}) + c(t - t_{_A}) because
F(t_{_A}) = A(t_{_A}))

When we see where point A is at the moment t_{_B} we obtain classically

A(t_{_B}) = A(t_{_A}) + v(t_{_B} - t_{_A})

Let us now calculate the distance between A(t_{_B}) and F(t_{_B}) at the moment t_{_B}. We write purely classically:

F(t_{_B}) - A(t_{_B}) = A(t_{_A}) + c(t_{_B} - t_{_A}) - A(t_{_A}) - v(t_{_B} - t_{_A}) = (c - v)(t_{_B} - t_{_A}),

However, when at time t_{_B} the photon reaches B the position of the photon and B coincide, therefore then F(t_{_B}) = B(t_{_B}). Also, B(t_{_B}) - A(t_{_B}) = r_{_AB} -- See here why ...

i.e., purely classically:

t_{_B} - t_{_A} = (B(t_{_B}) - A(t_{_B}))/(c - v) = r_{_AB}/(c - v),

or

t_{_B} - t_{_A} = r_{_AB}/(c - v),

The above expression is known from §2 of Einstein's 1905 paper, however, as seen above, it is derived purely classically and has nothing to do with STR!

It should also be noted that t_{_A} and t_{_B} are not the local times at A and B. For instance, let A be New York, NY and B be Kansas City, MO. The local time difference between New York and Kansas City, MO is one hour. If a photon starts at 2 o'clock pm local time in New York, NY it will end up in Kansas City, MO at 1 o'clock pm local time there. If t_{_B} - t_{_A} were the difference of the local times in New York and Kansas City, MO then it would mean that it would take an hour for a photon to travel from New York to Kansas City, MO. Of course, this is not so. A photon travels from New York to Kansas City, MO for a time of the order of 0.0001s. Therefore, obviously t_{_A} and t_{_B} are not the local times at A and B.