What is Einstein's equation? Or equations?

How many equations are there? We can represent a whole bunch of separate equations with a single tensor equation. There is just one tensor equation called "Einstein's equation", but when we break the tensor down into individual components, we get a whole bunch of individual equations. Thankfully, most of these are usually redundant, for reasons to be explained shortly.

First we'll define two tensors that we'll need to represent both sides of the set of equations.

In[2]:=

  
  DefineTensor[T,"T",{{2,1},1}]

  PermWeight::sym: Symmetries of T assigned

  PermWeight::def: Object T defined

In[3]:=

  
  DefineTensor[Reqn,"R",{{2,1},1}]

  PermWeight::sym: Symmetries of R assigned

  PermWeight::def: Object R defined

In[4]:=

  
  EinsteinG[la,lb]

Out[4]=

  G
   ab

In[5]:=

  
  Reqn[la_,lb_] := EinsteinG[la,lb] /. EinsteinToRicciRule

In[6]:=

  
  DefineTensor[eqn,"eqn",{{2,1},1}]

  PermWeight::sym: Symmetries of eqn assigned

  PermWeight::def: Object eqn defined

In[7]:=

  
  eqn[la_, lb_] := Reqn[la,lb] == k T[la,lb]

Now here it is! This is the Einstein equation, in tensor form, showing how spacetime curvature (the parts with R) is related to the distribution of energy (the part with T):

In[8]:=

  
  eqn[la,lb]

Out[8]=

  -(R g  )
       ab
  -------- + R   == k T
     2        ab       ab

The constant k is standing in for 8 Pi G/c^4, where G is Newton's gravitational constant, whose value measures the strength of the gravitational force, and c is the speed of light. Note that if we rescale the time coordinate by the speed of light c, that is like setting c=1, so we'll drop all the factors of c from now on with the understanding that all velocities are now measured relative to c.

Now let's look at the whole set of equations represented by the tensor equation in three spacetime dimensions. Notice that because we are working with symmetric tensors (T[1,2] = T[2,1]) the number of equations that we need to work with drops right away from nine to six.

In[9]:=

  
  Table[eqn[-i,-j],{i,3},{j,3}]

Out[9]=

    -(R g  )                 -(R g  )
         11                       21
  {{-------- + R   == k T  , -------- + R   == k T  , 
       2        11       11     2        21       21
   
     -(R g  )
          31
     -------- + R   == k T  }, 
        2        31       31
   
     -(R g  )                 -(R g  )
          21                       22
    {-------- + R   == k T  , -------- + R   == k T  , 
        2        21       21     2        22       22
   
     -(R g  )
          32
     -------- + R   == k T  }, 
        2        32       32
   
     -(R g  )                 -(R g  )
          31                       32
    {-------- + R   == k T  , -------- + R   == k T  , 
        2        31       31     2        32       32
   
     -(R g  )
          33
     -------- + R   == k T  }}
        2        33       33

As you will see below, only one of these equations will turn out to contain nontrivial information, and we'll only have to solve that one equation to understand everything we need to know about this spacetime.

Up to

What is the spacetime geometry?