Re: Length Of A Curve in Spacetime

On Mar 30, 5:09 pm, "Bronwyn" <BB@xxxxxxxx> wrote:
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

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

There is a problem here because the differential term is NOT dimensionless..
It has dimensions T/L and so the equation is dimensionally wrong.
The only way around this is to regard TIME as a fourth spatial dimension

Well, it's not THAT, but it can be measured in the same units of
space, even though it can't be treated just like a spatial dimension.

but
to do that would render just about all of modern science and technology
useless, which it obviously isn't.

No, not useless at all, just in the habit of doing something silly,
purely for historical reasons. If you want to get a feel for how silly
it is, I suggest you read the parable of the surveyors from the
opening chapter of Spacetime Physics, by Taylor and Wheeler. It's in
the library.

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

Meters. Or maybe seconds. Choose one. It's sorta like choosing inches
or centimeters.

PD

.

Relevant Pages

  • Re: Length Of A Curve in Spacetime
    ... I am confused about the way time and clocks are supposed ... different paths in spacetime. ... Note that dy/dx is a dimensionless ratio. ... meaning that "proper time runs slower than coordinate time", ...
    (sci.physics.relativity)
  • Re: Length Of A Curve in Spacetime
    ... Dirk Van de moortel wrote in message ... I am confused about the way time and clocks are supposed ... different paths in spacetime. ... Note that dy/dx is a dimensionless ratio. ...
    (sci.physics.relativity)
  • Re: Length Of A Curve in Spacetime
    ... I am confused about the way time and clocks are supposed ... different paths in spacetime. ... There is a problem here because the differential term is NOT dimensionless. ... the number of cycles ...
    (sci.physics.relativity)
  • Length Of A Curve in Spacetime
    ... 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. ... For a simple spatial plot like this: ... There is a problem here because the differential term is NOT dimensionless. ...
    (sci.physics.relativity)
  • Re: Length Of A Curve in Spacetime
    ... I am confused about the way time and clocks are ... Einstein wants it to be. ... | Note that dy/dx is a dimensionless ratio. ... Relativity has nothing to do with what everyone should know, ...
    (sci.physics.relativity)