[Phys-L] simultaneity at a distance.

[John Denker <jsd@av8n.com>]

Alas on 08/31/2016 09:00 AM, I wrote:

> Instead, consider the spacetime vector (Q-A) i.e. the interval from
> A to Q.  In some chosen frame F it has components [t, x]_@F.
>   -- Simultaneity in frame F means t_@F = 0.
>   -- In some other frame G, Q will be later than A, i.e. t_@G > 0.
>   -- In some other frame H, Q will be earlier than A, i.e. t_@H < 0.

QEWARE:  I did a bad thing.  I initially introduced A and B as
/different/ points separated by a /lightlike/ interval (i.e. null
interval).

As such, the t component cannot be zero in any frame.  If it were
zero, it would mean that the two points are not different.  It would
mean they are sitting on top of each other.

What I should have said is more like this:  Consider two *other* points
P and Q, separated by a *spacelike* interval.  Now consider the spacetime
vector (Q-P) i.e. the interval from P to Q.  In some chosen frame F
it has components [t, x]_@F.
>   -- Simultaneity in frame F means t_@F = 0.
>   -- In some other frame G, Q will be later than P, i.e. t_@G > 0.
>   -- In some other frame H, Q will be earlier than P, i.e. t_@H < 0.

=============

Returning to the original lightlike interval (B-A), if the t component
is positive in any frame, it will be positive in every frame.  There
is a sort of "triangle inequality" in spacetime that guarantees this.

The same is true for points separated by a /timelike/ interval.  For
such points, there is a well-defined invariant notion of "earlier"
and "later".

To say the same thing the other way:  The notion of time-ordering is
only a problem for /spacelike/ intervals.