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## Main Question or Discussion Point

From my reading of introductory texts on special relativity, I've seen this defined in various ways, and I'm curious about whether any of these definitions are preferable to others, for example because they're more convenient, consistent, logical, clearer, more widely used or more easily extended to general relativity. Most often, I've seen a Minkowski norm is defined as

[tex]\left||\mathbf{v}|\right| = \sqrt[+]{t^{2} - x^{2} - y^{2} - z^{2} }[/tex]

or

[tex]\left||\mathbf{v}|\right| = \sqrt[+]{-t^{2} + x^{2} + y^{2} + z^{2} }[/tex]

depending on the sign convention the author happens to be using. This "norm" fails to meet a number of the criteria of a true norm. It's not necessarily real valued (it can be real or imaginary), and the norm of a lightlike vector is always 0, even if the vector isn't

In

[tex]\left||\mathbf{v}|\right| = \sqrt[+]{\left|t^{2} - x^{2} - y^{2} - z^{2} \right|}[/tex]

ensuring that the value is always real and positive, although I think there's still the problem with lightlike vectors. Taylor & Wheeler in

[tex]\left(interval \right) = \left[ \left(\Delta t \right)^{2} - \left(\Delta x \right)^{2} \right]^{1/2},[/tex]

yet go on to define timelike and spacelike intervals (proper time and proper distance) in exactly the same way as Callahan defined his Minkowski norm:

[tex]\Delta \tau = \left[ \left(\Delta t \right)^{2} - \left(\Delta x \right)^{2} \right]^{1/2};[/tex]

[tex]\Delta \sigma = \left[ \left(\Delta x \right)^{2} - \left(\Delta t \right)^{2} \right]^{1/2}.[/tex]

They mention Minkowski's idea of multiplying the time coordinate by the square root of minus one,

http://elmer.tapir.caltech.edu/ph237/CourseOutlineA.html

says there are problems with extending this method to general relativity (adding that the concept of "imaginary time" which Stephen Hawking refers to in

[tex]\mathbf{e}_{0} \cdot \mathbf{e}_{0} = -1[/tex]

[tex]\mathbf{e}_{0} \cdot \mathbf{e}^{0} = 1.[/tex]

I've written in my notes: "The dot product is defined the same as for R

[tex]\left||\mathbf{a}| \right| = a^{i}a_{i}[/tex]

In response to a student's question, Kip Thorne says he won't be using the concept of the dual space in these lectures. If I've understood and correctly represented the ideas of this lecture in my notes, it seems that he's defined a kind of norm for Minkowski spacetime much more like the Euclidean norm that those other authors: always real, always positive except when zero... Does it fulfill all the requirements of a true norm? Does it sort of allow us to treat Minkowski spacetime as, in some ways, like R

[tex]g^{\alpha}_{\beta} = \left( \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)[/tex]

but not

[tex]g^{\alpha}^{\beta}[/tex] or [tex]g_{\alpha}_{\beta}[/tex]

defined by the same matrix. But would all of these be called different forms of the Lorentz metric (metric tensor)?

Finally, I see that the term Minkowski norm is also used for a rather different concept, synonymous with Hölder norm:

http://books.google.co.uk/books?id=PDjIV0iWa2cC&pg=PA17&lpg=PA17

the generalisation of the Euclidean norm which Wikipedia calls a p-norm:

http://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm

[tex]\left||\mathbf{v}|\right| = \sqrt[+]{t^{2} - x^{2} - y^{2} - z^{2} }[/tex]

or

[tex]\left||\mathbf{v}|\right| = \sqrt[+]{-t^{2} + x^{2} + y^{2} + z^{2} }[/tex]

depending on the sign convention the author happens to be using. This "norm" fails to meet a number of the criteria of a true norm. It's not necessarily real valued (it can be real or imaginary), and the norm of a lightlike vector is always 0, even if the vector isn't

**0**). I'm not sure about the triangle inequality...In

*The Geometry of Spacetime*, Callahan defines the Minkowski norm ||**v**|| of a vector**v**thus:[tex]\left||\mathbf{v}|\right| = \sqrt[+]{\left|t^{2} - x^{2} - y^{2} - z^{2} \right|}[/tex]

ensuring that the value is always real and positive, although I think there's still the problem with lightlike vectors. Taylor & Wheeler in

*Spacetime Physics*seem to fluctuate between these two approaches. They offer the following definition of "the interval":[tex]\left(interval \right) = \left[ \left(\Delta t \right)^{2} - \left(\Delta x \right)^{2} \right]^{1/2},[/tex]

yet go on to define timelike and spacelike intervals (proper time and proper distance) in exactly the same way as Callahan defined his Minkowski norm:

[tex]\Delta \tau = \left[ \left(\Delta t \right)^{2} - \left(\Delta x \right)^{2} \right]^{1/2};[/tex]

[tex]\Delta \sigma = \left[ \left(\Delta x \right)^{2} - \left(\Delta t \right)^{2} \right]^{1/2}.[/tex]

They mention Minkowski's idea of multiplying the time coordinate by the square root of minus one,

*i*, but dismiss it on the grounds that it serves no purpose except to obscure a genuine physical difference between time and space. Lawden does use this method in*An Introduction to Tensor Calculus, Relativity and Cosmology*, at least for the early chapters which deal with Cartesian tensors. But Kip Thorne in the*Introduction to general relativity*section (part II) of this lecture series (I can't remember which one exactly)http://elmer.tapir.caltech.edu/ph237/CourseOutlineA.html

says there are problems with extending this method to general relativity (adding that the concept of "imaginary time" which Stephen Hawking refers to in

*A Brief History of Time*is something else entirely). Instead he introduces an orthogonal basis for Minkowski spacetime where the spatial basis vectors each coincide with their corresponding reciprocal basis vectors, as in a Euclidean orthonormal basis, while the temporal basis vector is equal in length but opposite in sign to the corresponding reciprocal basis vector for time, so that[tex]\mathbf{e}_{0} \cdot \mathbf{e}_{0} = -1[/tex]

[tex]\mathbf{e}_{0} \cdot \mathbf{e}^{0} = 1.[/tex]

I've written in my notes: "The dot product is defined the same as for R

^{3}, and so is the length (the square root of the dot product of the vector with itself."[tex]\left||\mathbf{a}| \right| = a^{i}a_{i}[/tex]

In response to a student's question, Kip Thorne says he won't be using the concept of the dual space in these lectures. If I've understood and correctly represented the ideas of this lecture in my notes, it seems that he's defined a kind of norm for Minkowski spacetime much more like the Euclidean norm that those other authors: always real, always positive except when zero... Does it fulfill all the requirements of a true norm? Does it sort of allow us to treat Minkowski spacetime as, in some ways, like R

^{4}? Does it, in fact, effectively accomplish what multiplying the time coordinate by*i*was inteded to do? This "dot product" seems to be equivalent to[tex]g^{\alpha}_{\beta} = \left( \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right)[/tex]

but not

[tex]g^{\alpha}^{\beta}[/tex] or [tex]g_{\alpha}_{\beta}[/tex]

defined by the same matrix. But would all of these be called different forms of the Lorentz metric (metric tensor)?

Finally, I see that the term Minkowski norm is also used for a rather different concept, synonymous with Hölder norm:

http://books.google.co.uk/books?id=PDjIV0iWa2cC&pg=PA17&lpg=PA17

the generalisation of the Euclidean norm which Wikipedia calls a p-norm:

http://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm