- #1

Physicist97

- 31

- 4

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter Physicist97
- Start date

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

- #1

Physicist97

- 31

- 4

Physics news on Phys.org

- #2

- 6,724

- 431

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.

- #3

Physicist97

- 31

- 4

- #4

- 6,724

- 431

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).

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:##R=-{\kappa}g^{\mu\nu}T_{\mu\nu}##

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.

- #5

- 10,322

- 1,496

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.

- #6

m4r35n357

- 658

- 148

This looks like it might be the article you were referring to . . . ?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.

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.

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.

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.

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.

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.

- Replies
- 2

- Views
- 780

- Replies
- 8

- Views
- 521

- Replies
- 8

- Views
- 496

- Replies
- 4

- Views
- 678

- Replies
- 5

- Views
- 656

- Replies
- 5

- Views
- 2K

- Replies
- 17

- Views
- 2K

- Replies
- 4

- Views
- 3K

- Replies
- 6

- Views
- 1K

- Replies
- 30

- Views
- 759

Share: