To me, the most remarkable equation is (5):
I verified this by looking through the booklet, "Another Road to Schwarzschild", which Henrik kindly sent me:http://booksondemand.e-butik.se/?artnr=1129
When I tried to verify the above by following the logic given at the blog post:http://henrik77.wordpress.com/2011/04/1 ... arzschild/
I found myself trying to disprove it. But when I put in various situations it worked and the paper explains it well.
What I find surprising about this way of writing the acceleration is that it is written entirely in terms of the unit vectors
, the radius
, and the gamma factor related term
The strange thing about this term is that it appears to be very dependent on the direction of the velocity, but when one puts
, one finds that the result does not depend on the direction of velocity.
And I'm intrigued by this as this is the kind of thing I'd expect from a unification of gravity with the elementary particles.
That is, the elementary particles are defined in terms of left and right handed portions. These act very much different from each other. For example, only the left handed electron participates in the weak force. But the left and right handed portions correspond to particles moving with speed c. So a stationary electron has to be described as a combination of left and right handed portions moving at speed c in opposite directions. Thus the effect of gravity on such an object should depend on the direction in which it is pointed. But on the other hand, we know from observation that the gravitational force does not depend on direction of spin orientation.
So I'm going to see if I can rewrite the exact equations of motion for motion in the Schwarzschild metric into a form similar to the above. I'll start with the equations I published for the Schwarzschild metric in Gullstrand-Painleve metric, i.e. equation (10) of:http://arxiv.org/abs/0907.0660