The corridor example can be expanded in two dimensions, so that the corridor can be replaced by a triangle:
The first trianglet, corresponds to Einstein’s train example. Instead of a train, we can imagine an airplane, moving from point O to point A, emitting a photon towards point O΄, and taking back the photon at point A. Equivalently the description can be made with respect to an observer at rest, at point O΄, who emits a photon, and traces the airplane when the latter reaches point A. In such a sense the time Δt΄ is related to a clock on the airplane (proper time), while the time Δt is related to the clock of the observer at rest, at point O΄ (coordinate time).
Thus, with respect to the first triangle in the previous images, we have
which gives us the result we have already seen in Einstein’s train example.
The second triangle, is the two- dimensional analogue of the corridor example. In this case the airplane, as it leaves point O, emits a photon towards point O΄, and takes back the photon at point A. If T1 is the initial period of the photon at point O (before the plane starts to move), and T2 is the photon’s period when the airplane reaches point A, then we have
If the last equation is written in the following form,
it can be associated with the notion of length contraction in relativity.
However the interpretation here is fundamentally different. This is because the length contraction is not due to relative motion, but to the effects motion has on the photon, or on spacetime (if we treat photons as fluctuations of spacetime). Such effects will ultimately lead us to the principle of synchronicity, later on.
Notes:
There are two ways to treat the previous expression
One way is to associate the time Δt, which is the time the observer measures on his/her clock, with the initial period T1 of the reference photon (the period which the photon has before the observer starts to move). Thus setting,
we take,
Identifying now,
we have that,
Thus we take back the common relativistic expression.
On the other hand, if we associate the time Δt, with the final period T2 of the reference photon,
then we take,
This formula, with a positive sign in the square root, in fact gives us back the relativistic expression, as a linear approximation at small speeds.
The linear approximation of a function of the form (1+x)n, is given as
so that, setting x=(v2/c2), it will be
Thus, defining
where γL- is the common Lorentz factor,
and supposing that the speed v is sufficiently small, we take
Therefore the formula
will correspond to the relativistic expression
at the limit of small speeds, v<<c.
The significance is that the times t΄ and t, as measured by the clocks of observers (either moving, or at rest, respectively), can be intimately connected to the periods (T2 and T1, respectively) of the photon the observers use to make measurements.
This intricate relationship also reveals the deepest aspect that photons are oscillations in spacetime (if the oscillations are measured in the form of photons), so that the reduction of the period of the photons corresponds to the reduced period of the oscillating spacetime.
This period goes to zero, only if the speed of the moving observer becomes infinite, as it is suggested by the formula
where the altered period T2, becomes a submultiple n of the initial period T1.
As a consequence, faster than light travel is possible.
The first trianglet, corresponds to Einstein’s train example. Instead of a train, we can imagine an airplane, moving from point O to point A, emitting a photon towards point O΄, and taking back the photon at point A. Equivalently the description can be made with respect to an observer at rest, at point O΄, who emits a photon, and traces the airplane when the latter reaches point A. In such a sense the time Δt΄ is related to a clock on the airplane (proper time), while the time Δt is related to the clock of the observer at rest, at point O΄ (coordinate time).
Thus, with respect to the first triangle in the previous images, we have
The second triangle, is the two- dimensional analogue of the corridor example. In this case the airplane, as it leaves point O, emits a photon towards point O΄, and takes back the photon at point A. If T1 is the initial period of the photon at point O (before the plane starts to move), and T2 is the photon’s period when the airplane reaches point A, then we have
However the interpretation here is fundamentally different. This is because the length contraction is not due to relative motion, but to the effects motion has on the photon, or on spacetime (if we treat photons as fluctuations of spacetime). Such effects will ultimately lead us to the principle of synchronicity, later on.
Notes:
There are two ways to treat the previous expression
On the other hand, if we associate the time Δt, with the final period T2 of the reference photon,
The linear approximation of a function of the form (1+x)n, is given as
and supposing that the speed v is sufficiently small, we take
The significance is that the times t΄ and t, as measured by the clocks of observers (either moving, or at rest, respectively), can be intimately connected to the periods (T2 and T1, respectively) of the photon the observers use to make measurements.
This intricate relationship also reveals the deepest aspect that photons are oscillations in spacetime (if the oscillations are measured in the form of photons), so that the reduction of the period of the photons corresponds to the reduced period of the oscillating spacetime.
This period goes to zero, only if the speed of the moving observer becomes infinite, as it is suggested by the formula
As a consequence, faster than light travel is possible.
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