Combining space and time into four dimensions demonstrates invariance under Lorentz transformation.
Space-time being so fundamental an attribute of our universe, there must be numerous analogues. What invariances apply to them, if any?
This is a corollary to the question: https://www.researchgate.net/post/Apart_from_space-time_are_there_any_other_mathematical_models_of_physical_phenomena_using_a_fourth_dimension
And:
Preprint Size, scaling, and invariant ratios
No-global Lorentz invariance is one way of combining space and time (just like invariance under rotations is one way of combining the spatial dimensions).
The reason global Lorentz invariance is useful is that the wave equation is invariant under Lorentz transformations.
The generalization of global Lorentz invariance is invariance under general coordinate transformations-which describes general relativity; and the generalization of that is supergravity.
Robert Shour ,
The Lorentz transformations (LT), hence 4D space-time, given in the equivalent Lorentz' form is the following,
(1) t'= tγ-1 - vx'/c2
(2) x' = γ(x-vt)
where the primed and unprimed are relevant to two inertial frames.
The time transform (1) is a direct outcome of mixing a time dilation (experimental evidence, twin effect) and light-time variation
vx'/c2 = x'/c *v/c (additional time taken by light in reaching a moving target).
In accelerators the light-time x'/c is negligible due to the small dimension x' .
The tested relations become the following
(1) t'= tγ-1
(2) x' = γ(x-vt)
- it is simple time dilation and length contraction in 3D space and time.
- No test of Lorentz Invariance is performed in accelerators .
- if x'/c is large enough, LT do not reduce to Galilean transformations.
The LT have inbuilt the equivalence of inertial frames and the constancy of the one-way speed of light (SOL). Let's see what happens..
Experimental evidence in Radar Doppler complies to a good approximation with the formula, derivable from the LT
f/f0 = (1+β)/(1-β) where β = v/c
where f0 is the frequency of the radar and v is the speed of the target in one dimension.
Δf/f0 = 2β/(1-β) which gives,
being E0=hf0 it is ΔE/E0 = 2β/(1-β),
For an approaching target the frequency detected by the RADAR increases, so the energy in the inertial frame of the RADAR increases.
ΔE/E0 > 0.
Given two inertial frames (with mirrors) and transmitting light in the same way, by choosing one of the two, the net result would be that the energy measured in one frame at every bounce increases, and nothing else happens since the speed of the frames remains constant by definition of Inertial frame.
- This gives a perpetuum mobile of the first kind (something which Physics avoids like plague)
Since for half way or one way path it is simply ΔE/E0 = β/(1-β), no need to resort to a RADAR, the infringement occurs just by transmitting light from an inertial frame to another, some energy gets created.
This means that there cannot be an experiment involving light and two inertial frames to which real objects are attached, so the application of the equivalence of inertial frames which put them in relation, is not suitable for Physics.
The formula which gives the proper account for the conservation of energy and momentum (derived from it) is the following in agreement with quantum mechanics.
f/f0 = (1+β)/(1-β+ 2hf0 /mc2)
hence
Δf/f0 = 2(β-hf0 /mc2)/(1-β+ 2hf0 /mc2)
in the RADAR inertial frame where m is the mass of the object, the object is not inertial and the light as experimentally verified recoils (energy and momentum variation).
Conclusion: the LT involving light transmission (meaning always) cannot be applied in between inertial frames unless infringing conservation laws.
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