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bernhard.rothenstein
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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.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?
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
http://www.springerlink.com/content/n08028x00w00x516/
In the 3+1 case, certainly more than time dilation is needed.
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
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?
robphy said:What is the specific AJP paper you refer to?
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.
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.
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.
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.
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.