On Tuesday, December 19, 2017 at 3:34:41 AM UTC, Brent wrote: > > > > On 12/18/2017 6:54 PM, [email protected] <javascript:> wrote: > > > > On Tuesday, December 19, 2017 at 2:36:32 AM UTC, [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? Not >> clear from what you write. But whatever it is, why is that deemed to be >> invariant? Shouldn't it be the laws of physics, in this case gravity, and >> hence the field equations? AG * >> > > *I think you mean the dS^2 value is invariant along two paths with the > same endpoints, but not the path length of the spatial coordinates. > Correct? * > > > Right. > > *But why is this particular invariant so important, and perhaps used as a > guide to Einstein? AG* > > > Invariants are always the important things in physics because they are > what we can have intersubjective agreement on. > > Brent >
*IIUC, the field equations are covariant, which means coordinate system independent. Isn't Newton's Law of Gravitation also coordinate independent? That is, if we use Newton to calculate the planetary orbits, won't we get the same results in different coordinate systems? ... Is there a distinction in GR between frame independence and coordinate independence? AG* > > >> 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 >>> >> -- > 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] <javascript:>. > To post to this group, send email to [email protected] > <javascript:>. > Visit this group at https://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. > > > -- 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.
