Why Do Velocity & Momentum Equations Need Revision?

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Discussion Overview

The discussion centers around the need for revisions in the equations of velocity and momentum, particularly in the context of special relativity and Lorentz transformations. Participants explore the implications of tensor equations and the nature of vectors in different reference frames.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants argue that if a tensor equation is true in all reference frames, then equations involving velocity or momentum should also be true across frames, questioning the need for revisions.
  • Others clarify that while vector equations can be frame-independent, the coordinate representation of these vectors is not, and Lorentz transformations illustrate how these representations change.
  • A participant emphasizes that "true in all reference frames" means equality of vectors in different frames, not that a vector retains its form across frames, which relates to the misunderstanding of Lorentz transformations.
  • It is noted that 4-velocity and 4-momentum are valid in all Lorentz frames, contrasting with 3-vectors that depend on specific frames.
  • Another participant elaborates that 3-vectors are projections of 4-vectors onto a specific slice of spacetime, indicating that their components vary with the chosen frame of reference.
  • One participant discusses the distinction between non-relativistic and relativistic mechanics, explaining how the addition of vectors in flat spacetime differs from scenarios involving curved spacetime.

Areas of Agreement / Disagreement

Participants express differing views on the implications of tensor equations and the nature of vectors in various reference frames. There is no consensus on whether the need for revisions in velocity and momentum equations is justified, as multiple competing perspectives are presented.

Contextual Notes

Limitations include the dependence on definitions of vectors and tensors, as well as the unresolved nature of how these concepts apply in curved spacetime versus flat spacetime.

blumfeld0
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If a tensor equation is true in all reference frames and vectors are tensors of rank 1, then why aren't equations involcing velocity or momentum true in all reference frames?
why do they have to be revised, as, say, in special relativity ( the lorentz transformations)?


thanks
 
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If a vector equation is true in one reference frame then it is true in all inertial reference frames.

A + B = C, where A, B and C are all vectors, is a frame independent statement.

What is not frame independent is the coordinate representation of these vectors in a certain vector base. The Lorentz transformations show how these bases transform.

Garth
 
Last edited:
"True in all reference frames" means that if vector a= vector b in one reference frame then vector a= vector b in all reference frames. It does not mean that vector a in one reference frame = vector a in another reference frame which is what you appear to be thinking about when you refer to Lorentz transformations.
 
blumfeld0 said:
If a tensor equation is true in all reference frames and vectors are tensors of rank 1, then why aren't equations involcing velocity or momentum true in all reference frames?
why do they have to be revised, as, say, in special relativity ( the lorentz transformations)?
thanks
It is a 4-vector equation that is true in all Lorentz frames.
You seem to be referring to 3-vectors.
Equations involving 4-velocity or 4-momentum are true in all Lorentz frames
 
As we are dealing with Lorentz transformations then the vectors we are dealing with are 4-vectors, we are dealing with objects in both space and time.

A 3-vector is the projection of a 4-vector onto a particular foliation of space-time in a particular frame of reference.

These 3-vector components of a 4-vector are dependent on the particular 'slice' or foliation of space-time, which is dependent on the particular frame of reference.

Lorentz transformations convert from one particular foliation to another, that is from one observer to a second observer moving relative to the first.

Garth
 
Last edited:
blumfeld0 said:
If a tensor equation is true in all reference frames and vectors are tensors of rank 1, then why aren't equations involcing velocity or momentum true in all reference frames?
why do they have to be revised, as, say, in special relativity ( the lorentz transformations)?


thanks
velocity (= 3-velocity, i.e. the regular velocity that you know and love) and momentum (=3-momentum, i.e. regular momentum) are vectors in 3-dimensional space[i.e. the sum of all (x,y,z)]. I non-relativistic mechanics a (covariant) vector is anything whose components transform in the same way as the position vector r under an orthogonal transformation (i.e. rotations of axes) if and only if the space is flat (unlike the Earth's surface). In this way you can add 3-vectors with no problem. In relativity one speaks of 4-velocity and 4-momentum which are objects which lie in a 4-dimensional space aka spacetime [i.e. the sum of all (t, x, y, z)]. Thus if and only if the 4-dimensional space is flat you can add these vectors in a unique way. If the spacetime is curved then the resulting sum is not unique and will depend on how you "transport" one vector to the other which may be located at different places.

Pete
 
Thanks all for the great replies!

blumfeld0
 

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