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Einstein's Equation

This is the equation which makes General Relativity. In its various forms (and it has many) it describes the curvature of space-time, the excess radius of a sphere in that curved space-time and the rate of change of velocity of a ball of particles.

If we define our units nicely, we can bring Einstein's equation down to G_ab = T_ab. This is one of the most important equations in physics and is more frequently given as:

G_ab=8 pi k/c²T_ab Now that's lovely, but what does that tell us about gravity? Well, to find that out we first need something called the Einstein Tensor (which is that G thing on the left hand side), and then we're going to go through quite a complicated set of Tensor manouvres. The Einstein Tensor is as follows, using the Riemann tensor and Ricci scalar: G_ab = R_ab - (1/2)Rg_ab Don't have the faintest clue what that means, or why it's defined like that? You're not alone. We took it from John Baez's GR Tutorial , and he doesn't explain it much either. Just accept it for now. Now you've encountered this thing, how does it help us with the Einstein equation? Well, it appears on the left hand side, which means we can substitute in, giving us: R_ab - (1/2)Rg = T_ab (In our nice units, remember?) Now raising an index (which we described earlier) is a mathematical process, so we can apply it to both sides of an equation and get the same result. We do so here, and get: R^a_b - (1/2)Rg^a_b = T^a_b The exact same thing applies to contraction, so we can do that as well: R^a_a - (1/2)Rg^a_a = T^a_a Remember that we earlier defined the Ricci scalar, R, as being R^a_a and also g^a_a is 4 in four dimensions. Thus we finally end up with: R - 2R = T^a_a which, after solving reduces to: R = -T^a_a Great, another tensor equation! What does this mean? Well its actually very interesting as it states that the Ricci scalar equals the negative sum (because of our summation convention) of the diagonal components of the stress energy tensor. What are those? We found that they were the energy density at a point, and the momenta in the x, y, and z directions. Seems interesting, but what does that mean?

Now we have to do a bit of backtracking, and eventually get back to our Ricci tensor (you can skip this if you want). We start by putting our new formula for R back into the original Einstein equation, and then solving for R_ab:

R_ab + (1/2)T^c_cg_ab = T_ab;
R_ab = T_ab - (1/2)T^c_cg_ab Now, remember what we defined the Ricci tensor as earlier? If we take some particles, moving in the same direction with the same speed through space time, with the particle in the middle having velocity v and initial volume V, then -R_abv^av^b times V is the second time derivative of the ball's volume.

We can also change things around a bit in this second time derivative, so that we eventually get R_abv^av^b = R₀₀. This is extremely important, because it means we can substitute back into our equation, with the values a=b=0, and obtain:

R₀₀=T₀₀ + (1/2)T^c_c because in our Minkowski space time g₀₀ = -1. T₀₀ is the density of energy, but what is T^c_c? Reversing our contraction process, is g^caT_ac, where we sum over c and a. Our metric is the standard one, given here:

Which, applied, gives us -T₀₀ + T₁₁ + T₂₂ + T₃₃, and which after substituting into the last equation, gives us our final result of: R₀₀ = -(1/2)[T₀₀ + T₁₁ + T₂₂ + T₃₃] This result is the final one, and confirms most of our physical laws. The left hand side relates the rate at which a ball of particles converges, or their deviation, attraction, whatever you wish to call it, to the energy density of the ball as well as the sum of its pressures. We can use this to get curvature, gravitational fields, everything. This is the "geometrical essence" of General Relativity, related in this paragraph from John Baez's tutorial:

"Take any small ball of initially comoving test particles in free fall. As time passes, the rate at which the ball begins to shrink in volume is proportional to the energy density at the center of the ball plus the flow of x-momentum in the x direction, plus the flow of y-momentum in the y-direction, plus the flow of z-momentum in the z direction"

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