# Additiion of relativistic velocities. what do we need to derive it

• bernhard.rothenstein
In summary: Author claims that he derives the addition law of velocities based on time dilation and length contraction. my oppinion is that time dilation suffices. what is your oppinion?In summary, an Author claims that they derive the addition law of velocities based on time dilation and length contraction. My opinion is that using time dilation one can derive length contraction so all you really need is time dilation.
bernhard.rothenstein
in a recent paper in am.j.phys. an Author claims that he derives the addition law of velocities based on time dilation and length contraction. my oppinion is that time dilation suffices. what is your oppinion?

bernhard.rothenstein said:
in a recent paper in am.j.phys. an Author claims that he derives the addition law of velocities based on time dilation and length contraction. my oppinion is that time dilation suffices. what is your oppinion?
My opinion is that using time dilation one can derive length contraction so all you really need is time dilation.

Best wishes

Pete

In all of these various "what do I need to prove...", one really needs to spell out ALL of the assumptions. To say that one uses (say) "only time dilation" is probably insufficient. Are you making assumptions of isotropy? reflection symmetry? an underlying group property? etc... In other words, what aspects of Minkowski geometry are you using [explicitly or implicitly]? If you don't claim those aspects, are you suggesting that your "proof" works in a more general spacetime? Without CLEARLY specifying such assumptions, the proof of the proposed claim is not complete...

See the attachment on https://www.physicsforums.com/showpost.php?p=694535&postcount=8
From that, one could use other properties of Minkowski space to eliminate the explicit use of some other property.By the way,
for 1+1 Minkowski spacetime, Liebscher uses the cross-ratio (from projective geometry) to derive the composition of velocities

In the 3+1 case, certainly more than time dilation is needed.

Last edited:

robphy said:
In all of these various "what do I need to prove...", one really needs to spell out ALL of the assumptions. To say that one uses (say) "only time dilation" is probably insufficient. Are you making assumptions of isotropy? reflection symmetry? an underlying group property? etc... In other words, what aspects of Minkowski geometry are you using [explicitly or implicitly]? If you don't claim those aspects, are you suggesting that your "proof" works in a more general spacetime? Without CLEARLY specifying such assumptions, the proof of the proposed claim is not complete...

See the attachment on https://www.physicsforums.com/showpost.php?p=694535&postcount=8
From that, one could use other properties of Minkowski space to eliminate the explicit use of some other property.

By the way,
for 1+1 Minkowski spacetime, Liebscher uses the cross-ratio (from projective geometry) to derive the composition of velocities

In the 3+1 case, certainly more than time dilation is needed.

thank you very much for your competent help. as far as I know the derivation of time dilation, in its simplest variant, needs the invariance of the space-time interval i.e. all what that implies. Please let me know what supplementary facts does the 3+1 approach require (invariance of distances measured perpendicular to the direction of relative motion?
sine ira et studio

bernhard.rothenstein said:
thank you very much for your competent help. as far as I know the derivation of time dilation, in its simplest variant, needs the invariance of the space-time interval i.e. all what that implies. Please let me know what supplementary facts does the 3+1 approach require (invariance of distances measured perpendicular to the direction of relative motion?
sine ira et studio

As they say, "the devil is in the details"...
Until the details are spelled out [with precise mathematical definitions], it is not easy to say what further is minimally required [or what has implicitly or explicitly been assumed thus far]. The attachment I directed you to gives schematically, for example, a sampling of the numerous attempts to "derive the Lorentz Transformations". Along those lines, one should be able to find a similar path to the velocity-composition formula. Certainly, the full set of Lorentz Transformations is sufficient. But that is probably not what you want.

Are you looking for a "real proof"... something axiomatic?
Or a "pedagogical proof"... that is, a pedagogical plausibility argument?

robphy said:
As they say, "the devil is in the details"...
Until the details are spelled out [with precise mathematical definitions], it is not easy to say what further is minimally required [or what has implicitly or explicitly been assumed thus far]. The attachment I directed you to gives schematically, for example, a sampling of the numerous attempts to "derive the Lorentz Transformations". Along those lines, one should be able to find a similar path to the velocity-composition formula. Certainly, the full set of Lorentz Transformations is sufficient. But that is probably not what you want.

Are you looking for a "real proof"... something axiomatic?
Or a "pedagogical proof"... that is, a pedagogical plausibility argument?

a pedagogical plausability argument is what I am looking for
thanks

What is the specific AJP paper you refer to?

robphy said:
What is the specific AJP paper you refer to?

Relativistic velocity and acceleration transformations from thought experiments
W. N. Mathews, Jr.
Am. J. Phys. 73, 45 (2005)
That is
Thanks

Hi Bernard,

There is a difference between relativistic addition of velocities in one frame and relativistic transformation of velocities within one frame to within another. This is a common misspeaking in special relativity in the scientific field.

Any way, the known velocity relation corresponds to transformation. It can be easily and totally deduced from Lorentz Relations (Transformations), which carries all SR's features (unless for only one other thing).

Then, as far as I think in this thread, the question may be, better: Is time dilation enough for constituting Lorentz Transformations? Or, are we in need for Length Contraction also?!

Thanks to note the difference between: A fact used to deduce a theory and a fact that constitutes a theory.

Thanks and best regards.

Amr Morsi.

There is a difference between relativistic addition of velocities in one frame and relativistic transformation of velocities within one frame to within another. This is a common misspeaking in special relativity in the scientific field.
we should make a net difference between addition of velocities measured in a single reference frame and velocities measured in different reference frames. interesting enough in the first case we have relations like c+(-)u

Any way, the known velocity relation corresponds to transformation. It can be easily and totally deduced from Lorentz Relations (Transformations), which carries all SR's features (unless for only one other thing).
there are authors who consider that the use of the LET obscure the physics of the studied problem. The problem has much in common with the mathematics problem solved using simple mathematics or algebra. long time ago I have shown in am.j.phys. that we can derive the LET using the addition law of velocities. in some papers posted on arxiv i have shown, using a method proposed by Rosser, that the addition law of velocities leads to two relativistic identities which together with the concept of proper physical quantity leads to all transformation equations encountered in all chapters of physics. If not elegant it is time saving!

Then, as far as I think in this thread, the question may be, better: Is time dilation enough for constituting Lorentz Transformations? Or, are we in need for Length Contraction also?!
i referred to a paper published by Am.J.Phys. in which the author claims that in order to derive the addition of velocities he need length contraction and time dilation. my oppinion is that time dilation suffices. pmbphy confirmed my oppinion. The way in which you put the question is also interesting. I think that the most natural way to approach special relativity is to start with the two postulates, present Einstein's clock synchronization procedure and the way is paved for everything. A few learners are aware of the importance of clock synchronization in understanding special relativity.

Thanks to note the difference between: A fact used to deduce a theory and a fact that constitutes a theory.
the way in which you state the problem is subtle. do you mean that the the two postulates constitute a theory whereas a particular fact (facts?) can reveal its consequences ?
Thanks for giving me the opportunity to discuss with you some interesting problems helping me to improve the way in which I have stated the problem.
best regards.

Bernhard
the best things a physicist can offer to another one are information and constructive criticism

## 1. What is the formula for adding relativistic velocities?

The formula for adding relativistic velocities is V = (u + v) / (1 + uv/c^2), where V is the final velocity, u is the velocity of the object, v is the velocity of the observer, and c is the speed of light.

## 2. How is the formula derived?

The formula for adding relativistic velocities is derived using the principles of Special Relativity. It takes into account the effects of time dilation and length contraction, and is based on the Lorentz Transformation equations.

## 3. What is the significance of the speed of light in the formula?

The speed of light, c, is a fundamental constant in the formula for adding relativistic velocities. It represents the maximum speed at which any object can travel in the universe and is a key component in the theory of Special Relativity.

## 4. Can the formula be applied to any type of velocity?

Yes, the formula for adding relativistic velocities can be applied to any type of velocity, as long as it is measured relative to the speed of light. This includes velocities of objects, observers, and even light itself.

## 5. What are some real-world applications of this formula?

The formula for adding relativistic velocities has many real-world applications, including in the field of particle physics and space exploration. It is also used in the calculation of velocities in high-speed transportation systems, such as bullet trains and airplanes.

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