A Laymen's Guide To General Relativity: Part VII
November 30, 2015
Today's article is going to be a short one, because we are nearly at Einstein's equations of general relativity, with only one more minor tool needed to make the final leap.
In the last two articles I gave intuitive definitions of both the Riemann tensor and the Ricci tensor, which are used to define the curvature of spacetime. The third element of this trio of curvature measures is the Ricci scalar, which can be thought of as an average of the Ricci tensor, which itself is an average of the Riemann tensor. The Ricci scalar assigns each point in space and time a number, and that number gives the overall curvature of spacetime at that point.
Using the notation I introduced previously, in which the Ricci tensor is written as Ri(A,V), the Ricci scalar is defined as
In the last two articles I gave intuitive definitions of both the Riemann tensor and the Ricci tensor, which are used to define the curvature of spacetime. The third element of this trio of curvature measures is the Ricci scalar, which can be thought of as an average of the Ricci tensor, which itself is an average of the Riemann tensor. The Ricci scalar assigns each point in space and time a number, and that number gives the overall curvature of spacetime at that point.
Using the notation I introduced previously, in which the Ricci tensor is written as Ri(A,V), the Ricci scalar is defined as
Rc =g(x0,x0) Ri(x0,x0) + g(x1,x1) Ri(x1,x1) + g(x2,x2) Ri(x2,x2) + g(x3,x3) Ri(x3,x3)
where as before g(x,y) are the metric functions.
When combined with the definition of the Ricci tensor, the meaning of the Ricci scalar is just a sum over all of the combinations of basis vectors into parallelograms. Take basis vector xi, move it along xi and xj and then again along xj and xi, and then see how much the difference overlaps with vector xj. Now sum over all values of i and j. The end result is an average of the spacetime curvature over all of the possible planes (each plane is defined by two basis vectors).
That is the Ricci scalar, and it is the final piece needed to introduce Einstein's equations. And that is coming up next...
I am a theoretical physicist & mathematician, with training in electronics, programming, robotics, and a number of other related fields.
