On 12/18/2017 6:36 PM, [email protected] wrote:
On Monday, December 18, 2017 at 8:48:08 PM UTC, Brent wrote:
On 12/18/2017 12:19 AM, [email protected] <javascript:> wrote:
On Sunday, December 17, 2017 at 10:39:18 PM UTC,
[email protected] wrote:
On Sunday, December 17, 2017 at 12:21:27 AM UTC, Brent wrote:
On 12/16/2017 2:59 PM, [email protected] 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. 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.
Brent
The Lorentz transformation is just the simple limiting case of
flat, smooth spacetime. It's useful because in a sufficiently
small local region spacetime is going to be flat and smooth.
Brent
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