Bernardz wrote in message news:MPG.1a46d7ee77fd918c98979d@news...
In article ,
says...
Does anyone know of any references where
time is treated equally to the other 3Dimensions?

There are certain circumstances where a Wick rotation makes the
mathematics more tractible; a Wick rotation takes the real time
coordinate t into a purely imaginary coordinate s=it; in terms of
{x,y,z,s} the metric is diag(1,1,1,1). In some sense this treats s
("imaginary time") "the same" as space....


To some sense limited extent okay but is there any agreement on what
imaginary time could mean?

As far as I know, it's just a trick that's used to make an indefinite
metric positive definite. From what I've heard, there's alot more
understood about positive definite systems in mathematics than
indefinite systems, so it makes the theory easier to work with from a
pure math approach.

"Igor" wrote in message
om...
"Paul Riley" wrote in message
t...
One of the basic assumptions of GR is that "time" moves in one
direction.
Why treat time as a special case? Does anyone know of any references
where
time is treated equally to the other 3Dimensions?

None that would give you the proper interpretation of the physics.
One of the main motivations for using space-time coordinate systems in
the first place is to demonstrate the invariance of the speed of
light. This means that the square of the time coordinate must always
take the opposite sign of the square of the spatial coordinate when
determining the distance between two events in space-time. If they
both had the same signs, you could do it mathematically, but it just
wouldn't make any sense physically.

I understand that the relativistic quantities of for example Kinetic
Energy
is mc^2/sqrt(1-v^2/c^2) rather than 1/2mv^2


Yes it is. And the former is best approximated by mc^2 plus the
latter if we expand the denominator as a power series in v^2/c^2 and
keep only the first two terms.


But why is time considered a special dimension? Velocity is the
derivative
of distance with time, eg dx/dt

It's just that it has a different representation. In mathematical
terms, say it has a different signature. All squared space-time
intervals which are less than zero are said to be timelike, and those
greater than zero, spacelike. If it's exactly zero, we call it
lightlike since dx/dt = c.

Why not quantities dx/dy, dt/dy etc. where y is the preferred dimension?

Those indeed can have mathematical meaning, but usually not in the
context of mechanics, where we're more interested in rates of change.

What do the quantities of velocity and acceleration look like in 4D
where
all dimensions are treated equally?

In 4D, we always measure kinematic quantities such as velocity and
acceleration with respect to proper time interval ds = sqrt (dx^2 +
dy^2 + dz^2 - c^2 dt^2). We do this because this quantity is
invariant, unlike coordinate time t, which transforms between frames.
So the 4-velocity is given by (dx/ds, dy/ds, dz/ds, c dt/ds).
Similarly for the 4-acceleration, etc.

Getting clearer. but ds looks imaginary for small values of dx dy dz and
finite value of dt. should there be a minus sign within the sqrt?