Anamitra Palit

The paper has been reposted with some minor corrections:basic conclusions remain unchanged We have p1.p2>=m1m2 c^2 The relation v.v=c^2[gamma^2-1+1/gamma^2] remains unchanged Four Four 3.pdf

Markus Hanke has applied classical commonsense to interpret pseudo Riemannian geometry. when he talks of the projection issue Markus Hanke claims "This is meaningless. If you are projecting a shorter 4-vector onto a longer one, the result can never exceed the length of the longer vector, that’s just basic geometry." Classically if the norm of a vector is zero its components are individually zero. But this is not true of the null vector in pseudo Riemannian geometry.. Its norm is zero but the components are non zero. The null vector for the pseudo Riemann case,is parallel to itself and perpendicular to itself simultaneously[remember , the components are in zero in general].Can you prove mathematically the claim staked by Markus v1.v2<c^2? On the contrary the paper has derived mathematically v1.v2>=c^2; v1=v2 corresponds to the case v.v=c^2 have Next: It has been proved mathematically in the paper: Four dot product of momenta p1.p2>=m1 m2 c^4 Pl note (E/c)^2-|p|^2=m^2c^2 [implies E^2-|p|^2 c^2=m^2 c^4]. We do have from the above that is from (E/c)^2-|p|^2=m^2c^2, p.p=m^2c^2 In the paper we have derived E1E2-c^2|p1||p2|>=m1m2c^4 That again leads us to v.v=c^2[gamma^2-1+1/gamma^2] Formal consideration of photon momentum [as in standard theory] Now p.p=E^2-c^2|p|^2=m^2 c^4 Setting rest mass= zero on the right side only we obtain:E=pc [NB:On the left side of the last equation,with the rest mass tending to zero and gamma tending to infinity for the photon , the product m=m_0 gamma matches with the finite value on the right side] But from quantum mechanics: E=h nu=hc/lambda or pc=hc/lambda or p=h/lambda not equal to zero. [nu=frequency,h:Planck's constant, lambda=wavelength] For my derivation Markus thinks momentum of the photon = zero why? It is important to keep in the mind that the results derived in the paper have finally led to a discrepancy: v.v=c^2[gamma^2-1+1/gamma^2] It stands in contradiction to conventional results which require v.v=c^2. The very intention of the paper is to highlight this contradiction in the theory.

Various inconsistencies example equation (8) have been pointed out. [There is a typo in the next line;equation (42) has to be replaced by equation (8)] Each of the seven derivations in the paper reveals a discrepancy.

The Taylor series has been analyzed in the enclosed paper in several ways to reveal discrepancies. The analysis is of course of a mathematical nature. Requesting the attention of the audience to the mathematics of the paper and the issues ensuing from it.... Taylor1000.pdf

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