Moving emitters and absorbers of photons

Dipl.-Phys. Roland Sprenger, Germany 16.04.2024

Abstract

The “addition” respectively “subtraction” of velocities at moving emitters and absorbers of quanta at the speed of light is explained alternatively by means of a fifth dimension.

Emission

In the reference system of an observer the velocity of photons emitted by a body with relativistic speed always is the speed of light. This is regardless of whether body and photon have the same or opposite directions, that is if the velocities “add” or “subtract”.

The addition theorem of velocities u = (u´ + v) / (1 + (u´ v)/c²) (1)

for equal directions u´ = c results in u = (c + v) / (1 + (c v)/c²) = [(c + v) / (c + v)] ∙ c = c

and for opposite directions u´ = -c (with v ≠ c) u = (-c + v) / (1 + (-c v)/c²) = [(-c + v) / (c - v)] ∙ c = -c .

If the emitter moves in the positive direction and emits a photon in the coordinate origin a five-dimensional Minkowski-diagram [1] looks like figure 1.

Fig. 1: In the coordinate origin photons are emitted in the same and in the opposite direction of an emitter moving with v > 0 (0 < v < c).

For 0 < v < c results a turning angle φ < 90° [2] . The worldline of the emitter is the t´-axis (blue in the orthogonal projection). In the rest system of the emitter S´ the velocity of the forward emitted photon is u´ = c . So it moves there on the angle bisector (red, right side) which is angle bisector in the reference system of the observer S as well for any positive velocity of the emitter. So the velocity of the photon in S also is +c for all velocities +v of the emitter.

With u´= -c the photon moves opposite to the emitter, i.e. on the angle bisector above the negative x´-axis. This angle bisector has moved in a plane orthogonal to the framework S being rotated by the angle φ . As this photon in framework S´ moves at the speed -c on the appropriate angle bisector which is part of this orthogonal plane, it´s orthogonal projection to S is the angle bisector there too (red, left side). That means, that for the observer too the velocity of the photon always is -c .

Absorption

If a body absorbs Photons their velocity in it´s rest system always is the light speed, no matter if the emitter moves towards the absorber or opposite, i.e. if the velocities “add” or “subtract”. If the directions of emitter and photon are equal the speed v of the emitter be positive, if they are opposite negative, see fig. 2.

Fig. 2: Absorber A, Photon P and moving emitter E in the rest system of the absorber

From (1) follows for u´ = c

u = (c + v) / (1 + (c v/c²)) = (c + v) / ((c + v) / c) = c .

The result is independent of v, so of it´s sign too:

If v > 0 (0 < v < c) it´s the same case as shown in fig. 1 (right side) for equal directions. The framework of the observer then is the rest system of the absorber.

If v < 0 (-c < v < 0) the emitter moves to the left. Therefore it´s worldline in a Minkowski-diagram runs in the 2nd quadrant, above the negative part of the x-axis [1]. As it is the t´-axis as well (x´= const. = 0) and as the x´-axis because of the units s and Ls always is mirror-symmetrical to the angle bisector t = x (t´ = const. = 0) the x´-axis runs in the 4th quadrant, see fig. 3.

Fig. 3: An emitter moves away with v < 0 to the left (blue worldline) from the resting absorber and sends a photon to the right to the absorber.

The rotation angle φ for a rotation around the angle bisector w1 cannot be calculated with the formula φ=arccos⁡[ tan⁡(π/4-arctan⁡(v/c))] from [2], as for x > 1 arccos(x) is not defined. But the oblique-angled coordinate system S´ as well arises by a rotation of S around the angle bisector w2 by an angle ψ (see fig. 3). By this rotation the angle bisector w1 is turned into the 5th dimension too. At this the angle of 45° of the worldline of all photons, called lightline in [1], is preserved. The orthogonal projection of that lightline to system S is the angle bisector w1. So the velocity of the photon in question in the rest system of the absorber S is the lightspeed c even at an emitter which is moving away.

The angle ψ for v < 0 can be calculated with the formula

ψ = arccos ⁡[ tan⁡ (π/4 + arctan⁡ (v/c) ) ] .

Proof:

In paragraph 6 in [2] the formula tan ε = tan δ ∙ cos φ for the turning of an angle δ around one of its legs by angle φ and the orthogonal projection ε of the turned angle onto the original plane is proved. According to fig. 3 in this case the turned angle is the angle 45°, the rotation angle is the angle ψ and the projected angle is the angle β/2 . So the following applies:

tan (β/2) = tan 45° ∙ cos ψ ,

cos ψ = tan (β/2) .

According to fig. 3 applies β/2 = 45° - |α| , but because here always α < 0 , applies β/2 = 45° + α ,

and so cos ψ = tan (45° + α) . (2)

With tan α = v/c follows cos ψ = tan (π/4 + arctan (v/c))

or ψ = arccos [ tan (π/4 + arctan (v/c)) ] . (3)

From (2) follows with the addition theorem of the tangent

cos ψ = (1 + tan α) / (1 – tan α)

cos ψ = (1 + v/c) / (1 – v/c)

or ψ = arccos [ (1 + v/c) / (1 – v/c) ] (4)
(with v < 0) .

For oppositely equal velocities u and v , i.e. u = -v , follows from the point symmetry of the arctangent function

arctan (u/c) = -arctan (v/c)

and from that ψ(u) = φ(v) .

Oppositely equal velocities have equal rotation angles.

Summary

The paradoxical phenomena that the velocity of light always is the same although the emitter or the absorber move – and that even at opposite directions – become descriptively understandable within a five-dimensional spacetime. This is a further (see [2]) indication of the existence of a fifth dimension.

References

[1] Metzler Physik, J.B. Metzlersche Verlagsbuchhandlung, Stuttgart, 1988, S. 348

[2] Relativistic addition of velocities in a five-dimensional spacetime (roland-sprenger.de)