Dear Adrian Sfarti,

If the components of the vector are independent of the coordinates then the norm of the vector gives a geometrical structure different from the Minkowski structure.

Regards

Hi,

What do you mean by proper time?

The definition of Minkowski space is that the scalar norm  is invariant (constant)

Dear Adrian Sfarti,

Please excuse me for my reply which hurt you. I agree with you that I do not have the knowledge about the four vectors. I posted this question to remove my  misunderstanding about the  geometrical objects which may exist in the Minkowski  space -time.. I do not have any knowledge about a four vector, the norm of which is a scalar  function and independent of the proper time. Kindly explain me or give example if you feel that I can understand it otherwise, permit me to put my problem in more details so that it could be possible for me  to know about my confusion.

Regards                                 

Dear Adrian Sfarti,

Thank you very much for your reply. The two examples of the four vectors mentioned by you are sufficient to put my view about the existence of a scalar function in Minkowski space-time.

Since, I cannot write the mathematical language, I will try to explain my view with out the mathematical language to the extent I can.

1-The first example is the four momentum vector. The norm  of this vector is a constant including zero and hence, it is not a scalar function.

2-The second example is the four electromagnetic potential about which, I do not have any idea. Therefore, I will consider all  p a constossibilities. Let us define 'n' as the norm of the four electromagnetic potential. Here we have the following possibilities:

(a) If 'n' is a constant , it may not be treated as a scalar function.

(b) If 'n' is a variable, it will depend on the proper time 'tau' and hence, 'n' may be treated as a scalar function of the Minkowski space-time. The rate or change of the components of the electromagnetic potential with respect to 'n' are constants such that the forth constant  component may not be zero. The norm of this constant four vector is unity. The rate of change of 'n' with respect to proper time 'tau' will also. be not zero.

(c) Let us imagine that 'n' is a scalar function which is independent of the proper time 'tau'. Such scalar function either may not exits or if exits, it behaves like a constant because, the rate of change of 'n' with respect to proper time 'tau' is zero.

We would like to mention here that, in Minkowski space-time, there are four geometrical quantities. The coordinates system, the components of four velocity which are constants, the metric tensor the components of which are also constants and the proper time 'tau'. The geodesic equations of the geometrical structure imply that the coordinates system are expressed  as the constant four velocity multiplied by the proper time 'tau' and hence, in the Minkowski space-time there are only three geometrical objects to express a quantity These are the four velocity vector, the metric tensor and the proper time 'tau'. Therefore, the variation of a quantity(scalar,vector, tensor) will depend only on the proper time'tau'.

Kindly give your comments on the above especially on the part (c). I shall be highly thankful to you for the same.

Regards

Proper time itself is a Lorentz invariant quantity that is c2 (Δτ)2 = c2 (Δτ')2 where un-dashed and dashed coordinates refer to to different observers who are connected by Lorentz transformations. Here I have used the standard definition c2 (Δτ)2 = c2 (Δt)2 - (Δx)2. If you construct a mathematical function which is purely formed out of Lorentz invariant quantities it has to be necessarily Lorentz invariant. Sorry there is a typographical error in your question, is that what you had asked?

Make it massless; now it moves at the speed of light and has no proper time.

How about a constant scalar function?