On 12/19/2017 7:58 PM, agrayson2...@gmail.com wrote:
On Tuesday, December 19, 2017 at 8:58:18 PM UTC, Brent wrote:
On 12/18/2017 11:54 PM, agrays...@gmail.com <javascript:> wrote:
On Tuesday, December 19, 2017 at 3:32:22 AM UTC, Brent wrote:
On 12/18/2017 6:36 PM, agrays...@gmail.com wrote:
On Monday, December 18, 2017 at 8:48:08 PM UTC, Brent wrote:
On 12/18/2017 12:19 AM, agrays...@gmail.com wrote:
On Sunday, December 17, 2017 at 10:39:18 PM UTC,
agrays...@gmail.com wrote:
On Sunday, December 17, 2017 at 12:21:27 AM UTC,
Brent wrote:
On 12/16/2017 2:59 PM, agrays...@gmail.com wrote:
There's a problem applying SR in this
situation because neither the ground or
orbiting clock is an inertial frame.AG
An orbiting clock is in an inertial frame. An
inertial frame is just one in which no forces
are acting (and gravity is not a force) so that
it moves with constant momentum along a
geodesic. Although it's convenient for
engineering calculations, from a fundamental
veiwpoint there is no separate special
relativity and general relativity and no
separate clock corrections. General is just
special relativity in curved spacetime. So
clocks measure the 4-space interval along their
path - whether that path is geodesic (i.e.
inertial) or accelerated.
*Interesting way to look at it. So free falling in
a gravity field is an extension of SR. But the
thing I find puzzling is that in GR the curvature
of space-time is caused by the presence of mass,
yet I can draw the path of an accelerated body as
_necessarily_ a curve in a space-time diagram. I am
having trouble resolving these different sources of
curvature. AG*
*Einstein must have figured that since gravity produces
an acceleration field, and accelerating test particles
move along curved paths in space-time, he could replace
acceleration by inertial paths in a space-time curved
by the presence of mass-energy. But now, when comparing
test particles moving along different paths in
space-time, he couldn't use the Lorentz transformation
because the relative velocities of the frames are not
necessarily constant. So how did he propose to find the
correct transformation equations, and what are they?
And what were the laws of physics, in this case
gravity, that had to be invariant? AG*
What's invariant is the measure along a path in
spacetime - it's what an ideal clock measures. The
relation between the measure along two different paths
obviously depends on the lumpiness of the spacetime
through which they travel. It's as if I headed north
thru the Sierras while you sailed up the coast. There's
no simple relation between our path lengths even if we
travel between the same two points.
*So what's invariant along along two paths with the same
endpoints? *
It's not about two paths. The length of each path as
measured using Einstein's theory of the metric (i.e. as
warped by mass-energy) is an invariant. Just as the distance
your car's odometer would measure driving from NY to LA, it's
some number and it depends on (a) the path you took and (b)
the topography along that path. The interesting point is
that two such paths between a pair of events are different
durations as measured by clocks carried along the trips.
*Why is this surprising? If dS^2 is path invariant between two
space-time events, and dT^2 is time measured in the co-moving frame,
one would expect the time duration to be different along different
paths since the spatial length varies. *
It would surprise Newton. He assumed that the time interval between two
events was independent of a how a clock moved between them.
*AG*
That's contrary to Newton, for whom time was an invariant.
*Not clear from what you write. But whatever it is, why is
that deemed to be invariant? *
Because it doesn't depend on what reference system you use in
spacetime. It's measuring a distance which is a real thing,
not something relative/subjective.
*Shouldn't it be the laws of physics, in this case gravity,
and hence the field equations? AG *
It's the basis for them. They can be written in terms of an
extremal principle for the invariant path lengths.
*Is this the method Einstein used to derive the field equations? *
No, he worked from analogy with Newton's gravity potential.*
*
*But the field equations are not uniquely determined, so there must
have been additional guidelines. I read that Einstein tried many
different equations until he found the right set. And the set he
settled on in 1915 had been tried years earlier but mistakenly
rejected. AG*
Yes, I think that's true but I'm not up on the question.
Brent
*
*
*This is one of my key interests in this subject; to understand
the _method_ he used to derive the field equations. If so, why is
invariant path lengths such a crucial condition? I agree that
physics seeks invariants, but why this particular one? AG
*
Having an interval measure is obviously at the heart of a
geometric theory.
Brent
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