Only Minkowski or Galilei from Commutative Velocity Composition

In summary, the conversation discusses the derivation of the Lorentz transformation (LT) using the first postulate of special relativity (SR), assuming linearity and commutativity of velocity composition. The constant velocity v is defined using the transformation equation (1), and the inverse transformation is derived using the SR postulate 1. The velocity composition formula is calculated using equations (2) and (4), and assuming commutativity leads to the equation (6). By sorting equation (6) in a way that the left side depends only on v and the right side only on u, it is determined that the constant alpha must be equal to 1/c^2. This leads to the LT, which shows that the speed of light
  • #1
Sagittarius A-Star
Science Advisor
1,251
938
TL;DR Summary
Derive Lorentz transformation from SR postulate 1 (principle of relativity), assuming linearity and assuming, that velocity composition is commutative (if GT can be excluded)

In the paper "Nothing but Relativity" from Palash B. Pal, they use instead the group law, that two consecutive boosts in the same direction must yield again a boost (equations 24 and 25).
https://arxiv.org/abs/physics/0302045

Why does it follow from the group law, that velocity composition is commutative?
The LT can be derived from the first postulate of SR, assuming linearity an that velocity composition is commutative, and that GT can be excluded: ##t' \neq t##.

PF-LT2.png

Definition of the constant velocity ##v##:

##x' = 0 \Rightarrow x-vt=0\ \ \ \ \ \ ##(1)

With assumed linearity follows for the only possible transformation, that meets requirement (1), where ##A_v## may be a function of the constant velocity ##v##:

##\require{color} x' = \color{red}A_v(x-vt)\color{black}\ \ \ \ \ \ ##(2)

With SR postulate 1 (the laws of physics are the same in all inertial reference frames) follows, that the inverse transformation must have the same form, if the sign of ##v## is reversed:

##\require{color}x = A_v(\color{red}x'\color{black}+vt')\ \ \ \ \ \ ##(3)

Eliminating ##x'##, by plugging the right-hand side of equation (2) for ##\require{color} \color{red}x'\color{black}## into (3), and resolving (3) for ##t'## yields the transformation formula for time:

##t' = A_v(t-x\frac{1-\frac{1}{A_v^2}}{v})\ \ \ \ \ \ ##(4)

The velocity composition formula follows by calculating ##dx'/dt'## from equations (2) and (4), with ##u=dx/dt##:

##u' = dx'/dt' = \frac{A_v(dx-vdt)}{A_v(dt-dx\frac{1-1/A_v^2}{v})} = \frac{u-v}{1-u(1-1/A_v^2)/v}\ \ \ \ \ \ ##(5)

With assuming, that velocity composition is commutative, follows from (5):

##u \oplus (-v) = (-v) \oplus u##

##\frac{u-v}{1-u(1-1/A_v^2)/v} = \frac{(-v)+u}{1-v(1-1/A_u^2)/u} ##

## u(1-1/A_v^2)/v = v(1-1/A_u^2)/u \ \ \ \ \ \ ##(6)

Now I sort equation (6) in such a way, that the left side depends only on ##v## and the right side only on ##u##. This can only be, if both sides are constant (overall independent of the velocities).

##(1-1/A_v^2)/{v}^2 = (1-1/A_u^2)/{u}^2 := \alpha##
##\Rightarrow##
##A_v= \frac{1}{\sqrt{1-\alpha v^2}}\ \ \ \ \ \ ##(7)

Plugging the the right-hand side of (7) for ##A_v## into (2), (4) and (5) yields transformation formulas, now containing a yet to be determined constant ##\alpha##, that does not depend on the velocity ##v##:
$$x' = \frac{1}{\sqrt{1-\alpha v^2}} (x-vt)\ \ \ \ \ \ \text{(8)}$$
$$ t' = \frac{1}{\sqrt{1-\alpha v^2}}(t-vx \alpha)\ \ \ \ \ \ \text{(9)}$$
$$ u' = \frac{u-v}{1-uv \alpha}\ \ \ \ \ \ \text{(10)}$$
These are the only possible transformation formulas, which fulfill SR postulate 1 (the laws of physics are the same in all inertial reference frames), linearity and that velocity composition is commutative. SR postulate 2 (the vacuum speed of light is the same in all inertial frames) was not used.

Only one of the following three cases can be valid:
  1. ##\alpha < 0##
  2. ##\alpha = 0##
  3. ##\alpha > 0##
Case 1 can be excluded because of missing causality-invariance, see the linked paper "Nothing but Relativity".
Case 2, the GT, can be excluded when assuming ##t' \neq t##, which is the opposite of Newton's assumption of an "absolute time" (see equation 9).

Then, only case 3 can be valid. Equation (10) shows, that ##\alpha## must have as unit the inverse of the square of a velocity. Therefore, I can set

##\alpha := 1/c^2\ \ \ \ \ \ ##(11)

Then setting ##u := c## and ##\alpha := 1/c^2## in equation (10) shows, that ##c## is an invariant velocity. Experiments showed, that light moves with that invariant velocity.

Plugging the right-hand side of (11) for ##\alpha## into (8) and (9) yields the LT.

See also:
https://arxiv.org/abs/1504.02423

Why follows from the group law, that velocity composition is commutative?
 
Last edited:
  • Like
Likes Dale and vanhees71
Physics news on Phys.org

Related to Only Minkowski or Galilei from Commutative Velocity Composition

1. What is Minkowski-Galilei velocity composition?

Minkowski-Galilei velocity composition is a mathematical concept that describes how velocities combine in special relativity. It is based on the principles of Minkowski geometry and Galilean transformations.

2. What is the difference between Minkowski and Galilei composition?

The main difference between Minkowski and Galilei composition is that Minkowski composition takes into account the effects of time dilation and length contraction in special relativity, while Galilei composition does not. This means that Minkowski composition is more accurate for objects moving at high speeds.

3. Why is it called "Only Minkowski or Galilei" composition?

This term refers to the fact that there are only two possible ways to combine velocities in special relativity: either using Minkowski composition or Galilei composition. These two methods are mutually exclusive and cannot be used together.

4. How is Minkowski-Galilei velocity composition used in physics?

Minkowski-Galilei velocity composition is used to calculate the velocity of an object in special relativity. It is particularly useful in situations where objects are moving at high speeds, such as in particle accelerators or in space travel.

5. Are there any real-life applications of Minkowski-Galilei velocity composition?

Yes, Minkowski-Galilei velocity composition has many practical applications in physics and engineering. It is used in fields such as astrophysics, aerospace engineering, and particle physics to accurately calculate and predict the behavior of objects moving at high speeds.

Similar threads

  • Special and General Relativity
Replies
10
Views
681
  • Special and General Relativity
4
Replies
120
Views
6K
  • Special and General Relativity
Replies
22
Views
1K
Replies
14
Views
1K
  • Special and General Relativity
Replies
6
Views
1K
  • Special and General Relativity
Replies
5
Views
240
  • Special and General Relativity
Replies
18
Views
1K
  • Special and General Relativity
Replies
3
Views
1K
  • Special and General Relativity
Replies
3
Views
941
  • Special and General Relativity
5
Replies
146
Views
6K
Back
Top