The Einstein Field Equations

Features:

1. The image displays the Einstein field equations:

            8πG                 
     G_ij = ---  T_ij     Original form. 
             c4
                  
                     8πG
     G_ij + Λ*g_ij = ---  T_ij     Cosmological constant form.
                      c4

   The terms and factors are:
   1. The vacuum light speed c = 2.99792458*10**8 m/s (exact by definition) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns.
   2. The gravitational constant G = 6.67430(15)*10**(-11) (MKS units) is the same as for Newton's law of universal gravitation.
   3. The cosmological constant Λ (AKA Lambda) which Einstein introduced in 1917 for cosmology and used to obtain the STATIC Einstein universe (1917). Nowadays, it is used for the Λ-CDM model which is the standard model of cosmology (SMC, Λ-CDM model c.1995--). It gives the acceleration of the universe and is also used to give the large-scale structure of the universe when the highest level of accuracy is needed for that. It was a new hypothetical fundamental constant. Einstein did NOT initially call Λ the cosmological constant, but the name got attached to it later somewhere. Einstein called Λ the cosmological member in 1945 (Cormac O'Raifeartaigh, Historical and Philosophical Aspects of the Einstein World, 2019, p. 18).
      When Λ > 0 (which is the case for the Λ-CDM model), it causes a repulsion which could be considered at kind of anti-gravity, but its NOT what one ordinarily means by that term and no one calls it that. When Λ < 0, it causes an attraction, but NOT like gravity.
   4. The Einstein tensor G_ij = R_ij-(1/2)Rg_ij which is a tensor describing the geometry of spacetime. The Einstein tensor is constructed using the Ricci curvature tensor R_ij and the Ricci scalar curvature R.
   5. The metric tensor of general relativity g_ij.
   6. The energy-momentum tensor T_ij.
   References: CL-9, Wikipedia: Einstein field equations.

2. We will NOT explicate tensors here, but they are a compact way of writing a set of real numbers with a rather complex relationship to each other. But just for some insight, the tensors in the Einstein field equations can be represented by 4 X 4 matrices.

3. The Einstein field equations are, in fact, a set of differential equations---written very compactly---that are posited as true at every point in spacetime. Their solutions give the geometry of spacetime and this geometry is the manifestation of gravity in general relativity.

4. The general relativity geodesic equation tells mass-energy how to move in spacetime.

5. But, of course, mass-energy affects the geometry of spacetime via the Einstein field equations. So a real solution of geometry and motion for a system must be a simultaneous self-consistent solution of the Einstein field equations and the GR geodesic equation for all mass-energy in the system. This is hard and this why general relativity is hard to apply. To current knowledge circa 2020s, there are, in fact, only 6 top-level (i.e., general and important) exact solutions in general relativity (see Relativity file: general_relativity_exact_solutions.html).

   To explicate in a shorthand how the Einstein field equations and the GR geodesic equation interact, there is a mnemonic: the Mass-Curve-Motion Mnemonic:

   EOF

6. To expand a bit on the eplication above:
   1. In the Einstein field equations, mass-energy and momentum described by the energy-momentum tensor T_ij dictate the geometry of spacetime (described by the metric tensor of general relativity g_ij) which in general is non-Euclidean geometry.

   2. Now non-Euclidean geometry means among other things that one has curved space. It is very hard for humans to picture 3-dimensional curved space. But we can certainly picture 2-dimensional curved space as illustrated in the figure above (local link / general link: space_spherical.html). There are ways picturing 3-dimensional curved space as discussed and visualized in IAL 25: Black Holes: Picturing Curved Spaces.

   3. The curvature of space is how gravity manifests itself in general relativity. In other words, the geometry of spacetime is actually the gravitational structure of spacetime.

   4. Now since general relativity is only needed where for very strong gravity (like near black holes), for cosmology, and for very precise measurements of gravity effects (e.g., the perihelion shift of Mercury: IAL 25: The Perihelion Shift of Mercury below), it is clear that we seldom need to consider the curvature of space because it is very small relative to us. We are like microbes living on beach ball: their 2-dimensional curved space looks like 2-dimensional Euclidean space (i.e., flat space) to them---unless they do a circumnavigation.