"Perion" wrote in message ...

"Dirk Van de moortel" wrote
in message ...

Not so much an ad hoc manipulation, but rather a neat way
of abbreviating the meaningful (and apparently important)
physical quantity
- Dt^2 + Dx^2 + Dy^2 + Dz^2
to something that has physical meaning and that involves real
numbers only.

For two arbitrary events we could just as well have defined
S = - Dt^2 + Dx^2 + Dy^2 + Dz^2
and then considered 3 separate cases:
1) if S 0
- define Ds = sqrt(S)
- DON'T define D\tau
2) if S = 0
- define Ds = 0
- define D\tau = 0
3) if S 0
- define D\tau = sqrt(-S)
- DON'T define Ds

This way there would be no (potentially) imaginary numbers,
but it would require *every* subsequent calculation, proof or
problem to separately consider these 3 separate cases for any
possible pair of events in the problem at hand. That would be
extremely tedious and very silly since, through the well known
properties of the real and complex numbers, we perfectly know
that we would get the same results anyway.


I understand. It does simplify the situation. I'm curious why many authors
prefer the -1,1,1,1 signiture for the metric rather than 1,-1,-1,-1. Since
any possible observable event for is either timelike or null (lightlike) it
seems that it would be more natural for the time component in the interval
to be positive.

But the time component can be either positive or negative,
in both cases giving a real square.
I guess you mean that for observable events "it would be
more natural for the time component in the interval to be
real" and thus taking the (+---) signature. But that doesn't
hold either since with the (-+++) signature the definitions
are made the other way anyway ;-)

Then ds^2 = dt^2 - dx^2 - dy^2 - dz^2 which would always be
greater than or equal to zero..

Only for observable events as seen by a specific observer
who is present at one particuar event.
But there are other events as well. If you are an observer
then you can think about two observable events
(t,x,y,z) and (t+dt,x+dx,y+dy,z+dz)
while the second events can be observable or not by the
observer present at the first event. And even then you will
have to deal with unobservable events as well.

I know it makes no difference because then
they'd just define x_0 = ct, x_1 = -x, x_2 = -y, x_3 = -z, etc.

certainly not, since that would make no difference for the
squares.

Hmmm....
Never mind - I see why someone might prefer the former signiture.
[Sorry... I think I'm rambling]

perhaps a bit confused :-)
What really matters is that the invariant quantity is given a
name and a letter and it is convenient that physical quantities
are represented by real numbers. Whether one author uses
a quantity and another uses 'minus that quantity' or even
'10 times that quantity' does not matter.

Dirk Vdm