https://encyclopedia.thefreedictionary.com/Teleparallelism
So I just wonder why you say ##e_a## is orthonormal.
But although GR gave some examples, but no one has proved such a conclusion in mathematics, No one have proved that for all non-flat metric field all the corresponding compatible...
In the original definition of teleparalle connection, ##e_a## should be a holonomic frame(coordinate basis field), It is not necessarily to be orthonormal.
If you demand ##e_a## is orthonormal, and at the same time it is changing, then the basis field is obviously a non-coordinate basis field...
When you say the result from my calculation is wrong. then you should point out where I have made a mistake.
If you are serious , you must know that math is exactly the base of explaination of our world.
The difference between me and ergospherical is that he didn't demand the basis field is a coordinate basis, so they assert that there exist some extra terms between a teleparallel connection and Levi-Civita connection. and I demand that the basis should be coordinate basis field, then a...
Don't hurry,
if we start from the definition of Levi-Civita connection:
##\Gamma_{abc} = \frac{1}{2}(g_{ab,c} + g_{ac,b} - g_{bc,a})##
Then we can use the definition of metric field
##g_{ab} = {e_a}^I {e_b}^J \eta_{IJ}##
and we demand that ##e_a## is a coordinate basis field;
##[e_a , e_b]=0##...
Are you serious to say so? If my answers is obviously wrong, then it is very easy to point out where is my wrong. In fact the mathematical technics I used is very easy.
ergospherical in post #10 derived the compatibility between teleparallel connection and a metric field.
1. The teleparallel...
If you find that in my calculation I said if the basis field is a coordinate basis field, then it just means we only apply a coordinate transformation on metric space with ##\eta_{IJ}## , then isn't it natural that the cuavature of the Levi-Civita connection is zero?
Is there any book tell us that a metric field can also be compatible with zero-curvature teleparallel connection? How to get the teleparallel connection from a metric?
I only believe in calculations, if you point out the falses in my calculations, I'll be gald to accept it.
In fact all the...
But does it means in all cases they can not be equal? Is there anything wrong in my calculations?
And you tell me the partial derivative in metric compatible equation is 'a partial derivative with respect to the coordinates.'. then does it means tetrad ##e_a^J## should be a tetrad field?
If...
If the basis field ##e_a={e_a}^I e_I## is not a coordinate basis, then it means that:
##{c_{ab}}^c e_c = [e_a, e_b]\neq 0##
Using definition of metric:
##g_{ab} = {e_a}^I {e_b}^J \eta_{IJ}##
If we define the connection:
##\Gamma_{abc} = \frac{1}{2}(g_{ab,c} + g_{ac,b} - g_{bc,a} + c_{abc} +...
So ##e_a^J## should be a tetrad field. then from the equation:
##\partial_a {e_b}^{I} = {\Gamma^c}_{ab} {e_c}^{I}##
If ##e_a^J## is a tetrad field, then the connection can be written as:
## {\Gamma^c}_{ab}={e^c}_{I}\partial_a {e_b}^{I} ##
It is a teleparallel connection, and the curvature of...
Because compatible is defined with metric compatible equation, and this equation is different with the normal differential equation in ##IR^n##. But we can analyze the equation in such a way. At first we can write the metric compatible equation in the combination of two equations:
##\partial_a...