On 12/18/2017 11:54 PM, [email protected] wrote:
On Tuesday, December 19, 2017 at 3:32:22 AM UTC, Brent wrote:
On 12/18/2017 6:36 PM, [email protected] <javascript:> wrote:
On Monday, December 18, 2017 at 8:48:08 PM UTC, Brent wrote:
On 12/18/2017 12:19 AM, [email protected] 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.
*Is this the method Einstein used to derive the field equations? *
No, he worked from analogy with Newton's gravity potential.*
*
*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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