csmyth3025":1ugu5a3l said:
Many laymen like myself have read that Einstein, et al developed a set of ten non-linear partial differential equations to describe a point-like particle in a gravitational field (the Einstein Field Equations). Is it possible to explain, in layman's language, specifically what these equations tell us about such a particle - such as the direction that is "staight ahead" for the particle if it is in free-fall? What other types of information do these equations produce?
Chris
The geodesic equation is the one that tells "straight ahead" for a particle, or which gives the path of freely falling particles in curved spacetime. It's sorta seperate from the 10 mentioned above.
The coolest derivation of it that I've seen uses something called the minimal-coupling principle, or the comma-goes-to-semicolon rule. It's a rule for generalizing the familar equations we use in flat space, over into curved spacetime. Also, this derivation doesn't even touch the 10 non-linear equations you mentioned above.
Basically, in Newtonion mechanics, "freely falling" particles follow a straight line:
x= m*t + b ; m and b constants
if you differentiate that twice, you get
(d/dt)(d/dt)x = 0
This would normally be written in a different, and slightly more complex, format .... but ....
The minimal-coupling principle says to replace those regular derivatives with the covariant derivatives used in GR. Once you do that, out pops the Geodesic equation! So, it really is a generalization (in curved space) of straight lines in flat space.
Maybe not quite the laymen's description, but if you understand a little calculus, you can see how easily the geodesic equation comes about, and the description of "straight lines", or the shortest path, in curved space.
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