# Relativistic addition of velocities without lorentz transformations

I have studied
N.David Mermin "Relativistic addition of velocities directly from the constancy of the velocity of light," Am.J.Phys. 51 1130 1983 and others with the same subject quoted by the Author. He describes a derivation of the addition law that dispenses not only with the LT but also makes no us of length contraction, time dilation or the relativity of silmultaneity.
He starts directly using the concept of speed without mentioning how do we measure it absolute length/absolute time interval; proper length/coordinate time interval or proper length/poper time interval. Does that diminish the value of the derivation?

## Answers and Replies

bernhard,

This derivation is very nice to read.
Like for everything its value depends on its usage.

I think that the unity of SRT and the unity of GRT lie in the invariance with respect to coordinate transformations.
Therefore this derivations is mainly useful to show the consistency of the theory on one more example and to give -by training- additional intuition to the students.
However, too much use of this kind of special case illustrations could distract the students from the main point: invariance.

Personally I like to read such arguments as much as I like to forget them.
If we look back in the past all kind of geometrical construction used in geometry, mechanics and optics, we can see an analogy: all these constructions are very nice to learn but they can be forgotten since they do not represent the main aspect of these domains of mathematics or physics.

Michel

Meir Achuz
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I have studied
N.David Mermin "Relativistic addition of velocities directly from the constancy of the velocity of light," Am.J.Phys. 51 1130 1983 and others with the same subject quoted by the Author. He describes a derivation of the addition law that dispenses not only with the LT but also makes no us of length contraction, time dilation or the relativity of silmultaneity.
He starts directly using the concept of speed without mentioning how do we measure it absolute length/absolute time interval; proper length/coordinate time interval or proper length/poper time interval. Does that diminish the value of the derivation?
Actually, the addition of velocities can be an intermediate step in deriving the LT, and so come before the LT in the logic.

velocities in SR

Actually, the addition of velocities can be an intermediate step in deriving the LT, and so come before the LT in the logic.
Thanks. Probably I was not specific enough starting the thread. My problem is how the relative velocity V of the involved inertial reference frames appears in the relativistic formulas? Let us follow
Jay Orear, Physics 1979. (Ch.8.4 my book is in German).
He starts with the derivation of the time dilation formula using light clocks in relative motion with speed V without mentioning how we measure it.
The transformation equations of classical mechanics are presented as
x=x'+Vt' (1)
t=t' (2)
and the reader supposes that V is measured as abolute length/absolute time interval.
The guessed shape of the relativistic transformations is
x=Ax'+Bt' (3)
t=Dt'+Ex' (4)
considering that A,B,D and E are a function of the same V as in (1). Imposing different situations for which (3) and (4) should account the Lorentz transformations are derived
x=g(V)(x'+Vt') (5)
t=g(V)(t'+Vx'/cc) (6)
If (1) and (2) lead to
u=u'+V (7)
(5) and (6) leading to
u=(u'+V)/(1+Vu'/cc) (8)
(7) and (8) leading to the same result only for u'=0. Is that the result of the reciprocity: if you move relative to me I move relative to you with -V.
Could we say that not changing the notation for V when we go from classic to relativistic we act correctly? Is it worth to mention that fact when we present the subject to allert students? The textbooks I know do not mention that.

teachers and users of special relativity

bernhard,

This derivation is very nice to read.
Like for everything its value depends on its usage.

I think that the unity of SRT and the unity of GRT lie in the invariance with respect to coordinate transformations.
Therefore this derivations is mainly useful to show the consistency of the theory on one more example and to give -by training- additional intuition to the students.
However, too much use of this kind of special case illustrations could distract the students from the main point: invariance.

Personally I like to read such arguments as much as I like to forget them.
If we look back in the past all kind of geometrical construction used in geometry, mechanics and optics, we can see an analogy: all these constructions are very nice to learn but they can be forgotten since they do not represent the main aspect of these domains of mathematics or physics.

Michel
Michel,
thank you for the attention paid to my thread. As I see, it is a big difference between the way in which users and teachers of special relativity theory answer the questions raised by the participants on the Forum.
As a teacher of it I like the derivations which derive the formulas that account for the relativistic effects, injecting at a given point of the derivation the first principle showing finally that the Lorentz transformations account for all of them.
Bernhard