Lorentz transforms physical incoherence
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(Based on an example presented by Daryl McCullough)

We assume that a rocket is moving at speed v = 0.866 c
relative to the Earth, and we want to know what is
the relationship between the coordinates of an event,
in the frame of the Earth, and the coordinates of
the same event, in the frame of the rocket.

We assume that the rocket passes the Earth at an
event with Earth-coordinates tR=0, xR=0 and with
rocket-coordinates tR'=0, xR'=0.
(The primed variables are observed in the rocket
frame).

Let's pick an event E, in this case an explosion,
that takes place on a planet, at rest relatively
to the Earth. The planet is situated at a distance xE
form the Earth, and the explosion is observed after
a time interval tE, hence the Earth coordinates of
the event E are (xE, tE), and the corresponding
rocket coordinates are (xE', tE').

For simplicity, let's assume that the Earth, the
planet, and the rocket are all lined up, so that
we only need to consider the x-axis.

In this scenario, the distances are expressed
in light-years (ly), and the times in years.

After a time interval T, the rocket will be at
a distance v*T from the Earth, hence the distance
between the rocket and the event E is D = xE-vT.
In the rocket frame, the corresponding distance
is D' = D/gamma
(as v = 0.866 c, gamma = 1/sqrt(1-v^2/c^2) = 2).

Let xE = 3 ly and tE = 5 years.

Examples:

1) After a time interval T = xE/v, D = 0 and D' = 0.
Let's notice that T' = T/gamma. This is always the
case, as the ratio T/T' depends on the velocity of
the rocket, not on its position.

2) Let's take an arbitrary value T, for instance
T = 10 years.
From the formula D = xE -vT, we get
D = 3-0.866*10 = -5.66 ly, and
D' = D/gamma = -2.83 ly.
T' = T/gamma = 5 years.

3) Let's take T = 9.608 years. Then
D = 3-0.866*9.608 = -5.32 ly
D' = D/gamma = -5.32/2 = -2.66 ly
Of course, T' = 9.608/2 = 4.804 years.

Let's notice that those results have been obtained
by applying "Lorentzian" formulae.

Now let's use the Lorentz (or rather Einstein)
transformations:

t' = gamma(t-vx/c^2) and x' = gamma(x-vt)

Using the above notation, we get

tE' = gamma(tE-v*xE/c^2), thus
= 2(5-0.866*3) = 4.804 years

D' = gamma(xE-v*tE)
= gamma(3-0.866*5) = -2.66 ly,
which is exactly the value obtained in example 3,
by using T = tE'*gamma = 9.608 years.

Thus, gamma(xE-v*tE) = (xE-vT)/gamma.

But, in fact, *there is no physical relation between
time and position*. When the rocket travels an
arbitrary distance v*T wrt the Earth, T' is always
given by T/gamma, in other words, T' is independant
from v*T. If T = tE = 5 years, T' = 5/2 = 2.5 years,
not 4.804 years, and of course,
D = xE-v*tE = 3-0.866*5 = -1.33 ly, and
D'= D/gamma = -0.665 ly, not -2.66 ly.
(notice "en passant" that -0.665*gamma^2 = -2.66)

Let's now illustrate the physical incoherence
of the Lorentz time transformation
tE' = gamma(tE-v*xE/c^2).

We have mathematically seen above that
gamma (xE-v*tE) = (xE-v*T)/gamma, or
gamma^2 (v*tE - xE) = v*T - xE (1)

Let's tE = xE/v + delta t, thus
delta t = tE - xE/v
v*delta t = v*tE - xE
Replacing (v*tE - xE) by v*delta t in relation (1),
one gets
gamma^2 * v*delta t = v*T - xE
v*T = xE + gamma^2 * v*delta t
T = xE/v + gamma^2 * delta t,
T = xE/v + gamma^2 * (tE - xE/v) (2)

And indeed, with xE=3, tE=5 and gamma=2,
equation (2) leads to
T = 3/0.866 + 4(5 - 3/0.866)
= 3.464 + 4(5 - 3.464)
= 9.608 years

Equation (2) shows that the t' in the Lorentz time
transform t' = gamma(t-vx/c^2), expressed in the
Earth frame by T = t'*gamma, corresponds to the
time tau needed by the rocket to travel the
distance xE at the velocity v, increased by the
difference between the time coordinate tE of the
event E and the time tau, multiplied by the
factor gamma^2.

If the Lorentz time transformation were coherent,
the correction by gamma^2 would also apply to tau,
not only to the difference (tE-tau).
Indeed, there is no physical justification for
not treating homogenously the time coordinate tE.

Btw, such fancy correction by gamma^2 leads to
absurdities. For instance, when v is close to the
speed of light, the time corresponding to the
product gamma^2(tE-tau) can exceed the age of the
Universe!

The incoherence of the Lorenz time transform is
very obvious when the event coordinates are
(xE=0, tE=5). Then the transform gives
T = gamma^2 * tE = 4 * 5 = 20 years, instead of
T = 5 years!

Marcel Luttgens