Changes

::::: The word "relativity" dates from the early 1800s.  That's not what is being discussed here.  If preceded with "theory of" then there is no need to capitalize; if stand-alone, however, it does add clarification to capitalize as is done for other specific concepts that differ from the generic names.--[[User:Aschlafly|Andy Schlafly]] 23:52, 29 July 2010 (EDT)

::::: The word "relativity" dates from the early 1800s.  That's not what is being discussed here.  If preceded with "theory of" then there is no need to capitalize; if stand-alone, however, it does add clarification to capitalize as is done for other specific concepts that differ from the generic names.--[[User:Aschlafly|Andy Schlafly]] 23:52, 29 July 2010 (EDT)

== Curl of the gravitaional field ==

== Curl of the gravitational field ==

Sorry to get over-technical, but the fundamental law of "fictitious forces" (including gravity) is that the force field (divided by the mass of the test object) is

Sorry to get over-technical, but the fundamental law of "fictitious forces" (including gravity) is that the force field (divided by the mass of the test object) is

First, aren't the laws of physics supposed to be the same for everyone?  What made twin B's watch run slower?  Well, she ''knew'' she was traveling at high speed.  She brought an accelerometer with her in the rocket.  Just as twin B in the parking lot knew she was walking all over the place, turning around and such, twin B in space knew that her world-line was turning, and therefore wasn't a straight line (geodesic).  How did she know?  It takes force to make you deviate from a geodesic (this is really pretty much the same as Newton's laws of motion), and she felt the force.

First, aren't the laws of physics supposed to be the same for everyone?  What made twin B's watch run slower?  Well, she ''knew'' she was traveling at high speed.  She brought an accelerometer with her in the rocket.  Just as twin B in the parking lot knew she was walking all over the place, turning around and such, twin B in space knew that her world-line was turning, and therefore wasn't a straight line (geodesic).  How did she know?  It takes force to make you deviate from a geodesic (this is really pretty much the same as Newton's laws of motion), and she felt the force.

Second, how can we analyze the curvature of B's world line?  Here's where general relativity has to come in.  As soon as world lines start to curve, you have to measure their curvature, that is "geodesic curvature".  You get into complicated issues of curved coordinate systems (you're in one now; it's what you perceive as "gravity"!), and curved spacetime, and so on.  And you get into the <math>\Gamma\,</math> symbols, which measure the deviation from a geodesic, and hence the "fictitious forces" that you feel.  This is why general relativity is related to the "twin paradox", in that the space ship followed a curved trajectory and experienced acceleration.  But, to analyze the plain facts "twin paradox", all you really need to know is that twin B followed a crooked line.  Place your ruler on a diagonal on the graph of Minkowski space, draw the line out to Alpha Centauri.  Turn the ruler, draw the returning line.  Ignore the impossibly sharp corners.  Use special relativity to analyze the Lorentz transform for each section of B's world line.

Second, how can we analyze the curvature of B's world line?  Here's where general relativity has to come in.  As soon as world lines start to curve, you have to measure their curvature, that is "geodesic curvature".  You get into complicated issues of curved coordinate systems (you're in one now; it's what you perceive as "gravity"!), and curved spacetime, and so on.  And you get into the <math>\Gamma\,</math> symbols, which measure the deviation from a geodesic, and hence the "fictitious forces" that you feel.  This is why general relativity is related to the "twin paradox", in that the space ship followed a curved trajectory and experienced acceleration.  But, to analyze the plain facts of the "twin paradox", all you really need to know is that twin B followed a crooked line.  Place your ruler on a diagonal on the graph of Minkowski space, draw the line out to Alpha Centauri.  Turn the ruler, draw the returning line.  Ignore the impossibly sharp corners.  Use special relativity to analyze the Lorentz transform for each section of B's world line.

Oh, and to try to answer some of your specific questions, the gravitational field, under either Newtonian or relativistic mechanics, is a conservative field.  Its curl is zero.  If it weren't, conservation of energy would be violated, and we could make a perpetual motion machine by having a planet run around in circles picking up energy.  The "curl=0" aspect of gravity under general relativity is more complicated, because true vector fields have to be on Minkowski space, but it still conserves energy.  When Mercury orbits the Sun, its perihelion precesses because of relativistic effects, but its energy is conserved.

Oh, and to try to answer some of your specific questions, the gravitational field, under either Newtonian or relativistic mechanics, is a conservative field.  Its curl is zero.  If it weren't, conservation of energy would be violated, and we could make a perpetual motion machine by having a planet run around in circles picking up energy.  The "curl=0" aspect of gravity under general relativity is more complicated, because true vector fields have to be on Minkowski space, but it still conserves energy.  When Mercury orbits the Sun, its perihelion precesses because of relativistic effects, but its energy is conserved.