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

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Discussion Overview

The discussion revolves around the interpretation of a tensor calculation involving the angle between two vectors in Minkowski space, specifically addressing the implications of obtaining a cosine value greater than one. Participants explore the definitions of inner products, norms, and the meaning of angles in this context, as well as the nature of orthogonality in relation to time coordinates.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions the author's conclusion that the angle between the vectors is "not real" when the cosine value is -√2, seeking clarification on whether this means the angle has no meaning or is complex.
  • Another participant emphasizes the importance of the inner product in the calculation, suggesting that the definition of the inner product must consider the metric tensor in Minkowski space.
  • Some participants assert that the vectors in question are unit vectors, leading to a product of norms equal to one, which is relevant to the calculation of the inner product.
  • A participant raises the question of whether the identity for the dot product holds in Minkowski space, prompting further discussion on definitions and assumptions.
  • There is a suggestion that the situation may relate to a variant of the Cauchy-Schwarz inequality, although this remains speculative.
  • One participant expresses interest in understanding the concept of orthogonality concerning the time coordinate, noting that null vectors can be self-orthogonal.
  • Another participant mentions the existence of quadratic forms that allow for nonzero vectors to have a zero inner product, indicating a potential area for further exploration.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of the cosine value and its implications for angles in Minkowski space. There is no consensus on the meaning of "not real" angles or the validity of the definitions being used, indicating ongoing debate and exploration of the topic.

Contextual Notes

Limitations in the discussion include unresolved definitions of inner products and norms in the context of Minkowski space, as well as the implications of obtaining cosine values outside the conventional range. The nature of orthogonality and the characteristics of null vectors are also areas that require further clarification.

nomadreid
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TL;DR
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 = gijAiBi/√((AiAi)(BjBj)) (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?
 
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Let u and v be vectors then ##u\cdot v=|u||v|cosA##. Your calculation neglected }u|}v|.
 
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 AiBi=gijAiBj, and the square of the norm (length)of vector Ai is thus AiAi= gijAiAj. With the Minkowski metric, g11=g22=g33= -1, , g44=c2, and gij=0 for i≠j. This makes the two vectors unit vectors, and the calculation for the inner product is -√2.
 
|u||v| is the product of the norms, not the inner product.
 
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=gijAiAj=g11A1A1=1
B2=gijBiBj=g11B1B1+g44B4B4=1
 
I can't comment any further, since I am not familiar with this subject.
 
Does the identity ##u.v=|u||v|cos\theta ##
hold in Minkowski space?
 
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.
 
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If ##u\cdot v=|u||v|cos(\theta)##, how is ##cos(\theta) \gt 1##? It requires a weird definition for dot product.
 
  • #10
I guess it is a variant of the Cauchy-Schwarz inequality.
 
  • #11
IMHO it is not such an unusual definition (defined in post #3): after all, usually ds2=<dr, dr>= <eidxi,ejdxj> =<ei,ej>dxidxj=gijdxidxj, where <.,.> is the inner product, and ei 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.
 
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  • #12
nomadreid said:
IMHO it is not such an unusual definition (defined in post #3): after all, usually ds2=<dr, dr>= <eidxi,ejdxj> =<ei,ej>dxidxj=gijdxidxj, where <.,.> is the inner product, and ei 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.
 
  • #13
WWGD said:
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.
 
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  • #14
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.
 
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