Bondi ##k##-factor concept from first principles

[robphy]

A metric (or dot-product) where the eigenvector axes (along the future-forward and future-backward lightcone) are orthogonal $\vec u \cdot \vec v=0$ can't capture the causal character of the vectors with a Minkowski metric.

$\vec u +\vec v$ is a timelike vector in Minkowski spacetime, and
$\vec u -\vec v$ is a spacelike vector in Minkowski spacetime. (Think radar measurements.)

But
$(\vec u+\vec v)\cdot(\vec u+\vec v)=(\vec u \cdot \vec u)+(\vec v \cdot \vec v) +2 (\vec u \cdot \vec v)= (\vec u \cdot \vec u)+(\vec v \cdot \vec v)$
and
$(\vec u-\vec v)\cdot(\vec u-\vec v)=(\vec u \cdot \vec u)+(\vec v \cdot \vec v) -2 (\vec u \cdot \vec v)= (\vec u \cdot \vec u)+(\vec v \cdot \vec v)$
means that this metric (dot-product) doesn't distinguish timelike from spacelike.

From my "rotated graph paper" approach, the Minkowski metric for (1+1)-Minkowski spacetime seems to be captured by the signed area of the diamond for the vector along the diagonal of that diamond. This area (on this plane) does not rely on the dot-product of either Minkowski spacetime or Euclidean space.UPDATE:
One could also consider the eigenvectors of the Galilean transformation in (1+1)-Galilean spacetime…. the PHY 101 position-v-time graph.

Again, there are no timelike eigenvectors (in accord with the Relativity Principle).
There is an eigenvector which Galilean-null and Galilean-spacelike [absolute time] (using the degenerate temporal-metric $\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right)$ )
and its eigenvalue is 1 ["absolute length"].

As in the Minkowski case, I don’t think Euclidean geometry can capture the Galilean structure.

Spacetime geometry is a non-Euclidean geometry.

 

[tom.capizzi]

I'm trying to follow, but I'm unclear what timelike eigenvectors have to do with the Relativity Principle. Could you elaborate?

 

[robphy]

Inertial observers have future timelike 4-velocities. A timelike eigenvector would imply that one of those 4-velocities was special, violating the Relativity Principle. In special relativity, the lightlike vectors are eigenvectors, corresponding to an invariant speed, the speed of light.

 

[tom.capizzi]

I see your point, but I wonder, isn't that the whole point of Minkowski spacetime, that time is different from space? In any case, the discussion is focused on those light-like vectors. In the context of this being an isomorphism, the dot product is not the only way to determine causal character. The formula s² = c²t²-r² still distinguishes time-like and space-like from light-like. It equals (ct+r)(ct-r) = s/e^w u x s*e^w v = s² u x v = s². In this isomorphism, it is the cross-product which determines causal character, as well as the directed area of the invariant bivector, s² u x v. So, is the issue that there is no distinction using the dot product, or that the distinction is determined by the cross-product, instead?

 

[robphy]

I see your point, but I wonder, isn't that the whole point of Minkowski spacetime, that time is different from space? In any case, the discussion is focused on those light-like vectors. In the context of this being an isomorphism, the dot product is not the only way to determine causal character. The formula s² = c²t²-r² still distinguishes time-like and space-like from light-like. It equals (ct+r)(ct-r) = s/e^w u x s*e^w v = s² u x v = s². In this isomorphism, it is the cross-product which determines causal character, as well as the directed area of the invariant bivector, s² u x v. So, is the issue that there is no distinction using the dot product, or that the distinction is determined by the cross-product, instead?

The point of Minkowski spacetime is to explain and predict the results of experiments… not to force a structure that appeals to us.

My rotated graph paper exploits a property of areas in (1+1)-Minkowski spacetime and is in tune with radar methods and the Bondi k-calculus. In (2+1) and higher, the role of a metric now becomes important.

(In a 1-dim vector space one can compare vectors without a metric… but in 2dim, one needs a metric. Areas on a plane are like a 1dim vector space… to describes areas in space would require a metric.)

For a higher dimensions of Minkowski spacetime, I don’t have a formulation… so I don’t have anything to declare about the role of the cross-product in general.

I’m not sure where this conversation started from or where it’s going. But I think I have made my point about being aware of the eigenvectors of the boost transformation in the k-calculus, the meaning of no timelike eigenvectors as a statement of the relativity principle, and that the Euclidean metric is not sufficient to handle spacetime geometry. I make no other claims other than my geometrical construction with the diamonds and methods of counting calculation seem to capture many features of (1+1)-Minkowski spacetime using the Minkowski metric.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

An off topic subthread has been deleted. Thread reopened.