ABSTRACT: Assuming different infinities are unequal leads to strange and counter-intuitive mathematical results such as Ramanujan's ...
Monday, August 29, 2016
The New Field Equations
According to the theory of general relativity, mass causes spacetime to curve, and spacetime tells objects how to move. If you look at the equation below, it should be obvious why this is the case.
Well ... OK ... it is not obvious. Perhaps we can derive some field equations that are equivalent but more intuitive? Let's start with diagrams A and B below (r=radius; ct=spacetime portion of the radius; ct'=warped spacetime):
Diagram A shows an arbitrary sphere of space with no mass present. The field lines are straight (or flat) and connect the center with the outer edge. The field lines in diagram B curl like waves and pull all the space inward toward the center, shortening the spacetime wavelengths and creating a smaller sphere with more curvature. (To see more details on how this works, click here.)
The Lorentz equation above is pretty straight forward (G=Newton's constant). It shows how ct' is a function of mass (m). Add mass (m) and ct' shortens. We can use this equation as a model for our new field equations. Let's see what we can come up with:
The last equation above is kind of interesting. All the stuff on the left side must equal one. Let's multiply both sides by 8(pi)r, (the derivative of a sphere area) and do a few more steps (E=energy; T=energy density):
Add some Tensor indices:
We now have something equivalent to Einstein's field equations. Notice how each term contains an 8pi factor. We can do away with it.
What was not obvious before is now more obvious. If the stress-energy tensor (Tuv) changes, the spacetime variable (ct') also changes. Any particle in the vicinity will be affected by the changing ct', and move along a geodesic curve.
Notice there are a couple of 1/r^2's that can be factored. Time to dress this puppy up a little bit more:
We've come full circle. We now have a tensor version of the Lorentz factor we started with. To change things up a bit more, let's bring ct inside the parentheses.
Finally, we can name the ct and ct' tensors S and S', respectively:
Well ... OK ... it is not obvious. Perhaps we can derive some field equations that are equivalent but more intuitive? Let's start with diagrams A and B below (r=radius; ct=spacetime portion of the radius; ct'=warped spacetime):
Diagram A shows an arbitrary sphere of space with no mass present. The field lines are straight (or flat) and connect the center with the outer edge. The field lines in diagram B curl like waves and pull all the space inward toward the center, shortening the spacetime wavelengths and creating a smaller sphere with more curvature. (To see more details on how this works, click here.)
The Lorentz equation above is pretty straight forward (G=Newton's constant). It shows how ct' is a function of mass (m). Add mass (m) and ct' shortens. We can use this equation as a model for our new field equations. Let's see what we can come up with:
The last equation above is kind of interesting. All the stuff on the left side must equal one. Let's multiply both sides by 8(pi)r, (the derivative of a sphere area) and do a few more steps (E=energy; T=energy density):
Add some Tensor indices:
We now have something equivalent to Einstein's field equations. Notice how each term contains an 8pi factor. We can do away with it.
What was not obvious before is now more obvious. If the stress-energy tensor (Tuv) changes, the spacetime variable (ct') also changes. Any particle in the vicinity will be affected by the changing ct', and move along a geodesic curve.
Notice there are a couple of 1/r^2's that can be factored. Time to dress this puppy up a little bit more:
We've come full circle. We now have a tensor version of the Lorentz factor we started with. To change things up a bit more, let's bring ct inside the parentheses.
Finally, we can name the ct and ct' tensors S and S', respectively:
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