# Is This a Valid Simplification of Einstein's Equations?

• Physicist97
Einstein equations as##G_{ab} = 8 \pi T_{ab}##and then takes the trace to get the 1 equation##R = 8 \pi T##where the left-hand side is the Ricci scalar and the right-hand side is the trace of the stress-energy tensor. This equation is referred to as the "Einstein trace equation."In summary, when taking the trace of the Einstein field equations, you are left with only one equation instead of the full set of four equations. While this equation can provide some physical insight, it does not contain as much information as the original field equations and may not be as
Physicist97
Einstein's Equations are usually written ##R_{\mu\nu}-(1/2)g_{\mu\nu}R={\kappa}T_{\mu\nu}## , where ##\kappa## is a constant. If you were to multiply both sides by ##g^{\mu\nu}## , then this becomes ##R=-{\kappa}g^{\mu\nu}T_{\mu\nu}## . I have used the relations ##g^{\mu\nu}g_{\mu\nu}=4## and ##R=g^{\mu\nu}R_{\mu\nu}## . ##g^{\mu\nu}T_{\mu\nu}## is the trace of the Stress Energy Tensor, which now leaves you with ##R=-{\kappa}T^{\mu}_{\mu}## . It seems to me that writing the Einstein Field Equations this way simplifies them, but I've never seen them written this way. Have I done something wrong using this approach, or is it not practical to write them in this form? Thank you for any explanation :).

What you've done is to take the trace of both sides of the Einstein field equations. The result is valid, but it doesn't contain as much information as the original field equations. A metric could satisfy it without satisfying the field equations.

For example, if the stress-energy tensor is that of an electromagnetic field, then its trace is zero. Suppose that on the left-hand side we put the metric of Minkowski space. Then the trace equation is satisfied, but Minkowski space is not a solution of the field equations when there is an electromagnetic field present.

Physicist97
Would it then be better to write ##R=-{\kappa}g^{\mu\nu}T_{\mu\nu}## , ##g^{\mu\nu}R_{\mu\nu}=-{\kappa}g^{\mu\nu}T_{\mu\nu}## , ##R_{\mu\nu}=-{\kappa}T_{\mu\nu}## . I understand why the other equation didn't carry enough information. The Einstein Equations are actually 16 equations (10 due to symmetric tensors), and writing the trace takes away information, but this way you once again have 16 equations. Written as ##R_{\mu\nu}=-{\kappa}T_{\mu\nu}## would it carry the same information? Is it even right to write it like this?

Physicist97 said:
Written as ##R_{\mu\nu}=-{\kappa}T_{\mu\nu}## would it carry the same information? Is it even right to write it like this?

Written this way it isn't true. This would only be true in the case of a vacuum solution (right-hand side = 0).

Physicist97 said:
##R=-{\kappa}g^{\mu\nu}T_{\mu\nu}##
This could be more compactly written as ##R=-T^\mu{}_\mu##. (Let's take ##\kappa=1##.) The metric raises an index. This is a true statement about the trace, but doesn't carry as much information as the full field equations.

Physicist97 said:
##g^{\mu\nu}R_{\mu\nu}=-{\kappa}g^{\mu\nu}T_{\mu\nu}##

This is just a longer way of writing the preceding equation.

Physicist97
Physicist97 said:
Einstein's Equations are usually written ##R_{\mu\nu}-(1/2)g_{\mu\nu}R={\kappa}T_{\mu\nu}## , where ##\kappa## is a constant. If you were to multiply both sides by ##g^{\mu\nu}## , then this becomes ##R=-{\kappa}g^{\mu\nu}T_{\mu\nu}## . I have used the relations ##g^{\mu\nu}g_{\mu\nu}=4## and ##R=g^{\mu\nu}R_{\mu\nu}## .

It's certainly simpler, but you've lost some equations along the way - you are down to one equation after the contraction, wheras the full set has four equations. But Baez does basically this in some of his lecture notes on GR - for instance http://math.ucr.edu/home/baez/gr/outline2.html. I think there was another webpage by Baez which was en more similar than this, but I'm not quite sure which one.

The one equation you have left though does give a lot of physical insight, though, I think Baez discusses this as wll.

pervect said:
I think there was another webpage by Baez which was en more similar than this, but I'm not quite sure which one.

The one equation you have left though does give a lot of physical insight, though, I think Baez discusses this as wll.
This looks like it might be the article you were referring to . . . ?

## 1. What are Einstein's equations?

Einstein's equations, also known as the Einstein field equations, are the set of ten equations in Albert Einstein's theory of general relativity that describe the relationship between the curvature of space-time and the distribution of matter and energy in the universe.

## 2. Why is it important to simplify Einstein's equations?

Einstein's equations are complex and difficult to solve, so simplifying them can make them easier to work with and understand. This can help scientists make predictions and test the theory of general relativity in different scenarios.

## 3. What does it mean for a simplification of Einstein's equations to be valid?

A valid simplification of Einstein's equations means that the simplified version still accurately represents the relationship between space-time curvature and matter and energy distribution. This requires careful mathematical analysis and validation.

## 4. How are Einstein's equations typically simplified?

Einstein's equations can be simplified using a variety of techniques, including approximations, mathematical transformations, and simplifying assumptions. Each approach has its own advantages and limitations, and the choice of simplification method depends on the specific problem being studied.

## 5. What are some potential implications of a valid simplification of Einstein's equations?

A valid simplification of Einstein's equations could lead to a better understanding of the universe and its fundamental principles. It could also have practical applications, such as improving the accuracy of GPS systems or aiding in the development of future technologies like space travel.

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