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In special relativity, light rays in Minkowski spacetime $\mathbb{R}^n$ travel along the light cone which, by definition, consists of all null directions associated with an indefinite quadratic form $q(x) = x^TKx$. Find and sketch a picture of the light cone when the coefficient matrix K is

a.) $\pmatrix{1&0\\0&-1}$

b.) $\pmatrix{1&2\\2&3}$

c.) $\pmatrix{1&0&0\\0&-1&0\\0&0&-1}$

I don't even know where to begin. This chapter deals with positive definites and it gives this question.

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Do you know what "null direction" means? – Gerry Myerson Oct 8 '12 at 23:31
up vote 1 down vote accepted

Null vectors are, by definition, vectors of length zero. This length is measured by the norm associated with the quadratic form: $\sqrt{Q(v)}$.

So, null vectors will be the $x$ such that $x^T K x=0$.

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So, I will solve the matrices given and let them be homogenous, which then gives me the answer necessary to draw the light cone? – diimension Oct 9 '12 at 1:13
If by "let them be homogeneous" you mean "set $x^TKx$ equal to $0$", then I suppose so. Really, all of that is implicit in "solve $x^TKx=0$. – rschwieb Oct 9 '12 at 12:18

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