Anamitra Palit

The covariant derivative is indeed a tensor.The example in the attached paper considers the usual transformation of a rank two mixed tensor. And this leads to an equation revealing a discrepancy [equation (3) of the paper]. Thus standard theory has been used to project a discrepancy. Therefore something must be wrong with the standard theory itself.Otherwise why should the discrepancy be there?

The covariant derivative of a rank one contravariant tensor is a mixed tensor of rank two. Considering its transformation we arrive at a conflicting result[discrepancy] in the enclosed paper.Equation (3) should no be valid for an arbitrary tensor Transformation_Covariant_Derivative.pdf

What Ghideon has interpreted is not correct. x^2>=a^2 implies x>=|a| or x=<-|a|. In a similar vein we may write (a1 b1-a2 b2)^2>=(a1^2-a2^2)(b1^2-b2^2) implies a1 b1-a2 b2>=Sqrt[(a1^2-a2^2)(b1^2-b2^2] or a1 b1-a2 b2<=-Sqrt[(a1^2-a2^2)(b1^2-b2^2) One has to be careful about the 'or' connective.

Looking at the application there is no problem We have considered the signature(+,-,-,-) Indeed v.v=c^2 (v_t)^2- (v_x)^2-(v_y)^2-(v_z)^2=(c^2)v_t^2-|v|^2=c^2 Therefore (c^2)v_t^2-|v|^2]>0 The quantities under the square root sign ,as pointed out , will be positive.That aligns itself with the application. Thanking Ghideon for his comment.

Thanks for your advice/instruction In this article we have proved: 1) Four dot product of velocities v1.v2>=c^2 [equation (2) has been derived] 2) Four dot product of momenta p1.p2>=m1 m2 c^4 [Equation (6)] 3) Finally we have brought out a result:v.v=c^2[gamma^2-1+1/gamma^2] [eq (7)) which contradicts standard theory Some typos and remediable errors [of a minor nature though] have been attended to and corrected. The paper has been reposted below https://drive.google.com/file/d/1-fH2BcyO5QVljei90fq49mfNm0oKzYKp/view?usp=sharing Four Four.pdf

Certain issues regarding four velocity and four momentum have been discussed in the attached file. Requesting the audience to consider and address these issues.. Four Velocity 102.pdf