Re: Second Order Lorentz Transformations:Algebraically Verified


"symphony" <mathxxmaster@xxxxxxxxxxxxx> wrote in message news:1163110307.907809.320700@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Igor wrote:
symphony wrote:
The Beauty of Canonical Form
can be found in the Lorentz Transformations.
Now they are also available
in higher approximations!

Higher approximation to what? Whether or not your transformations are
valid, you cannot call them Lorentz Transformations, which are already
complete hyperbolic rotations.

With the Algebraic Verification of the Second Order Lorentz
Transformations,
nature can now be potentially better described by higher canonical
approximations.
And now invariant quantities that are also conserved
will soon be known for paths that are parabolically curved.
And for the observer who is uniformly accelerated,
his "proper time" can now be easily calculated.

Those statements don't make any sense since the physics will be
independent of the coordinate system anyway. His "proper time" won't
be any different than it ever was, since it must also be independent of
the coordinate system. Finding a new transformation won't affect the
underlying physics at all.

In ordinary experience (that is well explained by Newtonian Mechanics
[which is
good enough to navigate astronauts to and from outer space]) the
Galilean Transformations

X = x - vt (eq.1a)

T = t (eq.1b)

are valid for motion that involves observers who are
moving with constant relative velocity v. When the ratio v/c is (very)
small, the Lorentz Transformations reduce to these.

In ordinary experience, for motion involving observers who are
uniformly
accelerated with respect to each other, the counterparts to the above
are

X = x - (1/2)at^2 (eq.2a)

T = t (eq.2b)

where a denotes the constant acceleration.

When the (dimensionless) ratio a/L is (very) small then the Second
Order
Lorentz Transformations reduce to eqs.(2). And there is also an
invariance relation
that is a counterpart to the one in SR.

I have just begun to re - acquaint myself with GR. I am now looking for
the
equations that describe uniformly accelerated motion (which is only
"one
order of complexity" up from the constant velocities of SR) in that
theory.

Would one be correct in presuming, for the one spatial dimensional
case,
for conditions that approach those of flat space-time [where the
uniform acceleration could still be accomplished, perhaps, via rocket
power] that the GR equations would
reduce to the form of eqs. (2) which hold true in ordinary Newtonian
Mechanics (?)?


See also
http://hermes.physics.adelaide.edu.au/~dkoks/Faq/Relativity/SR/acceleration.html

Purely with special relativity:
http://users.telenet.be/vdmoortel/dirk/Physics/Acceleration.html
No need for general relativity to describe accelerated motion.

The equations at the end
x(t) = c^2/a ( sqrt( 1 + (a t/c)^2 ) -1 ) (eq 3a)
T(t) = c/a argsinh(a t/c) (eq 3b)
give the coordinate distance x covered by the accelerated object
and the time T elapsed on its clock, as a function of coordinate
time t for an object undergoing a constant proper acceleration a.

These reduce with a first (resp. zero) order Taylor approximation
around a = 0 ( i.e. for small a) to
x = 1/2 a t^2
T = t

So the coordinates of an arbitrary event (x,t) are given in the
accelerated frame as
X = x - 1/2 a t^2
T = t
which are your equations 2a and 2b.

Note that this breaks down for larger values of t, since the
third (resp. first) order approximations around a=0 of equations
3a and 3b will certainly involve higher powers of t as well.

Dirk Vdm


.

Relevant Pages