Re: Length Of A Curve in Spacetime

Bronwyn <BB@xxxxxxxx> wrote in message
mcUHj.4190$n8.3624@xxxxxxxxxxxxxxxxxxxxxxxxxx
I'm studying relativity at present and have been trying to follow some of
the messages here. I am confused about the way time and clocks are supposed
to vary with speed.
I have read that just as the odometers of two cars will read differently if
they take different routes between A and B, so will two clocks that take
different paths in spacetime. I cannot understand this.

As everyone should know, the length of a curve is given by the equation s =
integral(root[1+dy/dx)^2)dx]
Note that dy/dx is a dimensionless ratio.

For a simple spatial plot like this:

Y
*
* * *
* * *
*
*

__________________________X

where y is a known function of x, the answer is straightforward. Its units
are those of length.

However if a similar curve in a 2D plane of spacetime is represented by an
X-T graph:

t
*
* * *
* * *
*
*

__________________________X


Do you realize that you have an object going back in time here?

we now have the length as: s = integral(root[1+dt/dx)^2)dx]

That would be nothing more than the length of the curve on the
graph.


There is a problem here because the differential term is NOT dimensionless.
It has dimensions T/L and so the equation is dimensionally wrong.

Not if t is defined as t = <conventional clock time> * <light speed>
or if x is defined as
x = <conventional spatial coordinate> / <light speed>

The only way around this is to regard TIME as a fourth spatial dimension but
to do that would render just about all of modern science and technology
useless, which it obviously isn't.

Using a conversion factor for time... rendering just about all of
modern science and technology useless?

Can somebody please tell me what are the dimensions of a 'length' in
spacetime and what its units might be?

Usually one calculates the elapsed proper time
Delta(T') = int{ T1 to T2; sqrt( 1 - dx/dt(t) ) dt }

Note that for physical objects you have
dx/dt(t) < 1 everywhere,
so
Delta(T') < Delta(T)
meaning that "proper time runs slower than coordinate time",
and also that
Delta(T') > 0
meaning that "time always runs forward"

Also note that for a light signal you have
dx/dt(t) = 1 everywhere,
so
Delta(T') = 0
meaning that "no time elapses on a light signal".

You might be interested in reading the first 3 sections of
http://www.eftaylor.com/pub/chapter1.pdf

Dirk Vdm
.