On 3/6/2019 1:27 AM, [email protected] wrote:
On Wednesday, March 6, 2019 at 1:03:16 AM UTC-7, Brent wrote:
On 3/5/2019 10:02 PM, [email protected] <javascript:> wrote:
On Saturday, March 2, 2019 at 2:29:50 AM UTC-7,
[email protected] wrote:
On Friday, March 1, 2019 at 10:14:02 PM UTC-7,
[email protected] wrote:
On Thursday, February 28, 2019 at 12:09:27 PM UTC-7,
Brent wrote:
On 2/28/2019 4:07 AM, [email protected] wrote:
On Wednesday, February 27, 2019 at 8:10:16 PM UTC-7,
Brent wrote:
On 2/27/2019 4:58 PM, [email protected] wrote:
*Are you assuming uniqueness to tensors; that
only tensors can produce covariance in 4-space?
Is that established or a mathematical
speculation? TIA, AG *
That's looking at it the wrong way around.
Anything that transforms as an object in space,
must be representable by tensors. The informal
definition of a tensor is something that
transforms like an object, i.e. in three space
it's something that has a location and an
orientation and three extensions. Something that
doesn't transform as a tensor under coordinate
system changes is something that depends on the
arbitrary choice of coordinate system and so
cannot be a fundamental physical object.
Brent
1) Is it correct to say that tensors in E's field
equations can be represented as 4x4 matrices which
have different representations depending on the
coordinate system being used, but represent the same
object?
That's right as far as it goes. Tensors can be of
any order. The curvature tensor is 4x4x4x4.
2) In SR we use the LT to transform from
one*non-accelerating* frame to another. In GR, what
is the transformation for going from one
*accelerating* frame to another?
The Lorentz transform, but only in a local patch.
*That's what I thought you would say. But how does this
advance Einstein's presumed project of finding how the
laws of physics are invariant for accelerating frames?
How did it morph into a theory of gravity? TIA, AG *
*Or suppose, using GR, that two frames are NOT within the
same local patch. If we can't use the LT, how can we
transform from one frame to the other? TIA, AG *
*
*
*Or suppose we have two arbitrary accelerating frames, again
NOT within the same local patch, is it true that Maxwell's
Equations are covariant under some transformation, and what
is that transformation? TIA, AG*
*I think I can simplify my issue here, if indeed there is an
issue: did Einstein, or anyone, ever prove what I will call the
General Principle of Relativity, namely that the laws of physics
are invariant for accelerating frames? If the answer is
affirmative, is there a transformation equation for Maxwell's
Equations which leaves them unchanged for arbitrary accelerating
frames? TIA, AG
*
Your question isn't clear. If you're simply asking about the
equations describing physics/*as expressed*/ in an accelerating
(e.g. rotating) reference frame, that's pretty trivial. You write
the equations in whatever reference frame is convenient (usually
an inertial one) and then transform the coordinates to the
accelerated frame coordinates. But if you're asking about what
equations describe some physical system while it is being
accelerated as compared to it not being accelerated, that's more
complicated.
*Thanks, but I wasn't referring to either of those cases; rather, the
case of transforming from one accelerating frame to another
accelerating frame, and whether the laws of physics are invariant. *
For simplicity consider just flat Minkowski space time. If you know the
motion of a particle in reference frame, whether the reference frame is
accelerated or not, you can determine its motion in any other reference
frame. As for the particle path through spacetime, that's just some
geometric path and you're changing from describing it in one coordinate
system to describing it in another system...no physics is changing, just
the description. If the reference frames are accelerated you get extra
terms in this description, like "centrifugal acceleration" which are
just artifacts of the frame choice. This is the same as in Newtonian
mechanics.
But if the particle is actually accelerated, then there may be more to
the problem than just it's world line through spacetime. For example,
if the particle has an electric charge, then it will radiate when
accelerated and there will be a back reaction.
*Here the "laws" could be ME or Mechanics. It seem as if GR is a
special case for gravity, but I was asking whether invariance, or
covariance, has been generally established. *
Einstein's equations are written in a covariant form, so they look the
same for all (smooth) coordinate systems. But the problem arises on the
right hand side, the stress-energy tensor. If you are considering the
motion of a charged particle then the stress-energy tensor has to
include the EM field of the particle and the interaction of the particle
with that field. This requires a global spacetime solution since, due
to the curvature of spacetime, the particle can emit EM radiation at one
event and then run into that same radiation at a later event. The
solution may even include singularities; which we know are unphysical.
So there is no simple transformation between frames like the Lorentz
transform between inertial frames in flat spacetime. The classical
theory is probably not even self-consistent when applied globally.
Here's a paper that addresses a simple form of the problem:
https://arxiv.org/pdf/1509.08757.pdf
*Also, if the LT works locally in GR, how do we transform between
non-local frames?*
There can be no general answer to that. In curved spacetime, you would
have to solve the problem in a global frame to determine the relation
between two local frames. The LT can only be relied on within a local
patch where spacetime is approximately Minkowski.
Brent
*TIA, AG*
Maxwell's equations apply to the description of the EM field of an
accelerating charged particle and show that the particle loses
energy to an EM wave, but how the particle interacts with it's own
field when accelerated produces unrealistic results which were
superceded by quantum field theory. Bill Unruh showed that the
accelerated system interacts with the vacuum as though the vacuum
is hot.
Brent
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