- #1
Tempest
Suppose an atom that is at rest in an inertial reference frame simultaneously emits two identical photons in exactly the opposite direction. For all intent and purpose consider the radius of the atom as vanishingly small. Now consider things after one second has passed in this 'atomic frame'.
According to the theory of SR, any photon in an inertial reference frame must have speed c=299792458 meters per second in that frame. Thus, after one second has passed, the distance each photon is away from the atom, in the atomic frame absolutely has to be 299792458 meters. Let us denote this distance by D.
Now consider things from the point of view of a coordinate system whose origin is one of the photons. In this system, the speed of this photon is zero, and the atom is moving (lets say to the left) at speed c, and the other photon is moving to the left at a speed greater than c.
Now, either the distance the atom is away from the photon in this photonic frame has been Lorentz contracted or not. Assume it has been contracted. Let d denote the distance the atom and photon are separated by in the photonic frame, after 1 second has passed in the atomic frame.
The exact amount of contraction is given by:
d = D(1-v^2/c^2)^1/2 = D(1-c^2/c^2)^1/2 = 0
Hence, after one second has passed in the atomic frame, the distance between the two photons in the photonic frame is zero. Hence, in the photonic frame, the two photons are not in relative motion, which is impossible (that would only be true if they were emitted in the same direction, and they were emitted in opposite directions). Thus, the Galilean transformations hold when we switch from the atomic frame to the photonic frame.
Now, consider a third reference frame F3, which was initially moving away from the atom (to the right) at a constant speed S, and such that after one second had passed in the atomic frame, photon 1 was located at the origin of F3.
We know that in the atomic frame, that after one second has passed photon one traveled 299792458 meters. Suppose that S = 1 meter per second. Thus, F3 is moving away from the atom at a constant speed of one meter per second.
And after one second has passed in the atomic frame, we have stipulated that the photon is at the origin of F3. Therefore, at the moment the photon's were emitted, the origin of F3 was located 299792457 meters away from the atom, in the atom's frame. It would then follow that after one second had passed in the atomic frame, that the origin of F3 would be 299792458 meters away from the atom, and hence the photon would be at the origin of F3 as stipulated, so everything is fine.
Hence in the atomic frame (atom at rest), we have an event which begins when the origin of F3 is 299792457 meters away from the atom, and ends when the origin of F3 is 299792458 meters away from the atom, and this event takes 1 second in the atomic frame.
(work in progress)
According to the theory of SR, any photon in an inertial reference frame must have speed c=299792458 meters per second in that frame. Thus, after one second has passed, the distance each photon is away from the atom, in the atomic frame absolutely has to be 299792458 meters. Let us denote this distance by D.
Now consider things from the point of view of a coordinate system whose origin is one of the photons. In this system, the speed of this photon is zero, and the atom is moving (lets say to the left) at speed c, and the other photon is moving to the left at a speed greater than c.
Now, either the distance the atom is away from the photon in this photonic frame has been Lorentz contracted or not. Assume it has been contracted. Let d denote the distance the atom and photon are separated by in the photonic frame, after 1 second has passed in the atomic frame.
The exact amount of contraction is given by:
d = D(1-v^2/c^2)^1/2 = D(1-c^2/c^2)^1/2 = 0
Hence, after one second has passed in the atomic frame, the distance between the two photons in the photonic frame is zero. Hence, in the photonic frame, the two photons are not in relative motion, which is impossible (that would only be true if they were emitted in the same direction, and they were emitted in opposite directions). Thus, the Galilean transformations hold when we switch from the atomic frame to the photonic frame.
Now, consider a third reference frame F3, which was initially moving away from the atom (to the right) at a constant speed S, and such that after one second had passed in the atomic frame, photon 1 was located at the origin of F3.
We know that in the atomic frame, that after one second has passed photon one traveled 299792458 meters. Suppose that S = 1 meter per second. Thus, F3 is moving away from the atom at a constant speed of one meter per second.
And after one second has passed in the atomic frame, we have stipulated that the photon is at the origin of F3. Therefore, at the moment the photon's were emitted, the origin of F3 was located 299792457 meters away from the atom, in the atom's frame. It would then follow that after one second had passed in the atomic frame, that the origin of F3 would be 299792458 meters away from the atom, and hence the photon would be at the origin of F3 as stipulated, so everything is fine.
Hence in the atomic frame (atom at rest), we have an event which begins when the origin of F3 is 299792457 meters away from the atom, and ends when the origin of F3 is 299792458 meters away from the atom, and this event takes 1 second in the atomic frame.
(work in progress)