Tensor calculation, giving|cos A|>1: how to interpret

[nomadreid]

TL;DR Summary

In the Minkowski metric, using the usual inner product relationship, finding the angle A between (1,0,0,0) and (2^0.5, 0, 0, (3^0.5)/c gives cos A = -2^0.5, which indicates that A is not real, but what is it? (source given)

On pages 42-43 of the book "Tensors: Mathematics of Differential Geometry and Relativity" by Zafar Ahsan (Delhi, 2018), the calculation for the angle between Aⁱ=(1,0,0,0) (the superscript being tensor, not exponent, notation) and Bⁱ=(√2,0,0,(√3)/c), where c is the speed of light, in the Minkowski metric with the (- - - +) convention, i.e.,
ds²=-dx²-dy²-dz²+c²⋅dt²,
is carried out with the calculation cos A = g_ijAⁱBⁱ/√((A_iAⁱ)(B_jB^j)) (Einstein summation convention). The result, -√2, then tells us that |cos A| = √2, which the author uses to conclude: "the angle between Aⁱ and Bⁱ is not real." I do not understand this: does he mean that the concept of angle has no meaning here, or that the angle is complex (whatever that means), or what?

 

[mathman]

Let u and v be vectors then $u\cdot v=|u||v|cosA$. Your calculation neglected }u|}v|.

 

[nomadreid]

By }u|}v| I presume you mean the inner product. But the calculation is less straightforward because we are in curved space. The author defines the inner product between the vectors as AⁱB_i=g_ijAⁱB^j, and the square of the norm (length)of vector Aⁱ is thus AⁱA_i= g_ijAⁱA^j. With the Minkowski metric, g₁₁=g₂₂=g₃₃= -1, , g₄₄=c², and g_ij=0 for i≠j. This makes the two vectors unit vectors, and the calculation for the inner product is -√2.

 

[mathman]

|u||v| is the product of the norms, not the inner product.

 

[nomadreid]

Ah, OK. I did not forget the product of the norms in the calculation; as I pointed out, these are (in this metric) unit vectors, so the product of the norms is 1:
A²=g_ijAⁱA^j=g₁₁A¹A¹=1
B²=g_ijBⁱB^j=g₁₁B¹B¹+g₄₄B⁴B⁴=1

 

[mathman]

I can't comment any further, since I am not familiar with this subject.

 

[WWGD]

Does the identity $u.v=|u||v|cos\theta$
hold in Minkowski space?

 

[nomadreid]

WWGD, the author of the book I am citing assumes that the identity holds in Minkowski space (and other metric spaces), with the definitions of inner product and norm taking into account the metric tensor in the manner I cited in my other posts in this thread. But he also allows for the angle to be "not real", giving a cosine value outside of that usually defined for cosine, which is what puzzles me, and which prompted my original question.

 

[mathman]

If $u\cdot v=|u||v|cos(\theta)$, how is $cos(\theta) \gt 1$? It requires a weird definition for dot product.

 

[WWGD]

I guess it is a variant of the Cauchy-Schwarz inequality.

 

[nomadreid]

IMHO it is not such an unusual definition (defined in post #3): after all, usually ds²=<dr, dr>= <e_idxⁱ,e_jdx^j> =<e_i,e_j>dxⁱdx^j=g_ijdxⁱdx^j, where <.,.> is the inner product, and e_i is a coordinate basis vector. So, the definition of the inner product as given seems to be natural, and fulfills all the requirements of an inner product. I am glad to be corrected, of course.

 

[WWGD]

IMHO it is not such an unusual definition (defined in post #3): after all, usually ds²=<dr, dr>= <e_idxⁱ,e_jdx^j> =<e_i,e_j>dxⁱdx^j=g_ijdxⁱdx^j, where <.,.> is the inner product, and e_i is a coordinate basis vector. So, the definition of the inner product as given seems to be natural, and fulfills all the requirements of an inner product. I am glad to be corrected, of course.

I would be interested in understanding the meaning of orthogonality regarding the time coordinate.

 

[nomadreid]

I would be interested in understanding the meaning of orthogonality regarding the time coordinate.

The key point to go on would be, I suppose, the definition of two vectors being orthogonal if their inner product (as defined in my previous post) is equal to zero. One remark out of this source (which I need to work out an example for) is that a null vector, that is, one with magnitude zero, need not be the zero vector (which means that null vectors are self-orthogonal, which sounds rather paradoxical). To complete the connection with relativity, the author defines vectors with real magnitudes as "time-like" and those with imaginary magnitudes as "space-like." Intuition flew out the window some time ago.

 

[WWGD]

There are quadratic forms Q ( obviously not standard inner products) for which there are nonzero a with
Q(a,a)=0. I will look for examples.