I'm not sure what prompted you to change the speed of light depending on the motion of the participant or the observer.
Do you have solid experimental data supporting your ideas?
Light is an electromagnetic wave, which means that its behaviour follows straight from the rules of electromagnetism. The speed of light is just a property that emerges from the relationship between electric and magnetic fields; there's no free parameters that you can fiddle with to speed it up and slow it down including its frequency. A light wave moving any faster or slower than 𝑐 is a light wave that doesn't obey Maxwell's equations.
There are two ways that one could perhaps deal with this. You could keep light behaving like a classical moving object, and have the laws of electromagnetism change depending on your speed (this might be a little troubling; you are held together by electromagnetism). Or you could keep the laws of electromagnetism the same, and instead alter the transformation that happens when you change velocity, having lengths contract and times dilate in just the right way to keep light moving at 𝑐.
Nature does the second one, and the resultant transformation is the Lorentz transformation you know and love.
Newtonian transformations, set of equations in classical
physics that relate the
space and time coordinates of two systems moving at a constant
velocity relative to each other. Adequate to describe phenomena at speeds much smaller than the
speed of light, Galilean
transformations formally express the ideas that space and time are absolute; that length, time, and mass are independent of the relative
motion of the observer; and that the speed of light depends upon the relative motion of the observer.
Lorentz transformations are the
set of equations in relativity
physics that relate the space and time coordinates of two systems moving at a constant
velocity relative to each other. Required to describe high-speed phenomena approaching the
speed of light, Lorentz
transformations formally express the relativity concepts that space and time are not absolute; that length, time, and mass depend on the relative
motion of the observer; and that the speed of light in a vacuum is constant and independent of the motion of the observer or the source. The equations were developed by the Dutch physicist
Hendrik Antoon Lorentz in 1904.
Lorentz transformation is the relationship between two different coordinate frames that move at a constant velocity and are relative to each other. The name of the transformation comes from Dutch physicist Hendrik Lorentz.
There are two frames of reference, which are:
- Inertial Frames – Motion with a constant velocity
- Non-Inertial Frames – Rotational motion with constant angular velocity, acceleration in curved paths
Lorentz transformation is only related to change in the inertial frames, usually in the context of special relativity. This transformation is a type of linear transformation in which mapping occurs between 2 modules that include vector spaces. In linear transformation, the operations of scalar multiplication and additions are preserved. This transformation has a number of instinctive features, such as the observer that is moving at different velocities may measure elapsed times, different distances, and ordering of events but the condition that needs to be followed is that the speed of light should be the same in all the inertial frames.
Lorentz transformation can also include rotation of space, a rotation that is free of this transformation is called Lorentz Boost. The space-time interval which occurs between any two events is preserved by this transformation.
To derive the Lorentz Transformations, we will again consider two inertial observers, moving with respect to each other at a velocity v. This is illustrated in Figure 1. This time, we will refer to the coordinates of the train-bound observer with primed quantities. We will assume that the two observers have synchronized their clocks so that t = t′ = 0, and at this point in time, the two origins coincide. We will also assume that the two coordinate systems move with respect to each other along the x-direction - in particular, the ground-based observer witnesses the train moving in the positive x-direction. We now imagine that some event occurs inside of the train (or anywhere else in space, for that matter), which is described by the train-bound observer using the coordinates (x′, y′, z′, t′). A common example might be that a firecracker explodes at some location in space, at some point in time. How do these coordinates relate to the ones that the ground-based observer would use to describe the event, (x, y, z, t)?
When two inertial observers experience the same event:
First, we notice that we must have
y′ = y ; z′ = z.
This is because lengths perpendicular to the direction of motion cannot contract without leading to a variety of physical paradoxes. To understand why, imagine that as the train is sitting still at a station on the ground, its height is just barely taller than a tunnel which is further down the track. After the train leaves the station and is moving at some speed v, if the ground-based observer were to witness a contraction of the train’s height, he might believe that the train is now shorter than the tunnel, and can make it through the tunnel without
crashing. However, the train-based observer will believe that the tunnel is now even shorter than it was before, and still cannot accommodate the train. Thus, the two observers disagree as to whether the train does or does not crash, which is nonsensical.
We do know, however, that the direction parallel to the motion will contract. Now, the coordinate x′ is the distance between the location of the event, as measured by the train-based observer, and the origin O′ that the train-based observer has set up for his coordinates. This same distance, according to the ground-based observer, is x − vt. The reason for this is that according to the ground-based observer, the distance between the origins O and O′ is vt, while the distance to the event is simply x. This is illustrated in Figure 1. Therefore, we have two competing expressions, according to the two different observers, for the physical distance between O′ and the spatial location of the event. Since we know that physical distances will contract according to the length contraction formula, we find
x − vt = x′/γ
Thus, if we know the time and location of an event in the coordinates of the ground-based observer, we know the location of the event according to the train- based observer.
As for the time coordinate, we can employ a clever trick based on the above result. Since both observers are inertial, and the situation between them is entirely symmetric, we should be able to consider the inverse transformation for the position coordinate. Since this simply amounts to swapping the roles of the primed and unprimed coordinates, along with sending v → −v (since the train-based observer witnesses the ground moving in the opposite direction), similar considerations lead to the result
x=γ(x′ +vt′)
If we now substitute in our result for x′, and solve for t′, the result we find is
t=γt−vx/c .
Therefore, if we know the coordinates of an event as described by the ground- based observer, we know that we can find the coordinates of the event as de- scribed by the train-based observer, according to the formulas
x′ =γ(x−vt) y′ = y
z′ = z
t′ =γt−vx/c2
These expressions together are known as a Lorentz transformation. While we have derived them for a specific orientation of the two coordinate systems, deriving them in the more general case is straight-forward (although unnecessary for our purposes). Again, we can find the inverse transformation simply by swapping the roles of the coordinates, and sending v → −v,
reduces to
x′ = x−vt y′ = y
z′ = z
t′ = t
x = γ ( x ′ + v t ′ )
y = y′ ′
z=z
t = γ t′ + vx′/c2
Thus, the small-velocity limit of the Lorentz transformation is the Galilean transformation, which of course it must be. For hundreds of years, it was widely believed that the Galilean transformation was correct, because according to every experiment ever conducted, it was correct.
If Special relativity is to be a correct theory of nature, it must explain the outcomes of all experiments, including these ones. The fact that the Lorentz transformation reduces to the Galilean one in this limit is proof that Special Relativity can account for those experiments, ones which were of course conducted long before any physicists knew anything about the postulates of special relativity.
By arguing that any two events which are causally connected must have an absolutely unambiguous time ordering, we have found that only events within the light cone of the origin are capable of having a causal influence on it. However, by the definition of the light cone, this is just the statement that causal information cannot propagate faster than the speed of light, since any point which lies outside of the light cone of the origin would need to communicate with the origin via a signal which propagated faster than c. Thus, we again arrive at our conclusion that no causal influence can propagate faster than the speed of light.
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