On Tuesday, December 19, 2017 at 8:58:18 PM UTC, Brent wrote: > > > > On 12/18/2017 11:54 PM, [email protected] <javascript:> wrote: > > > > On Tuesday, December 19, 2017 at 3:32:22 AM UTC, Brent wrote: >> >> >> >> 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] 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. >> > *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. 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* > > > *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 > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
