Components of Einstein's Equations in 4 dimensions

In summary, these equations represent the 3 spatial coordinates and 1 time coordinate, which are the only coordinates that are truly independent. The other 2 components represent unphysical degrees of freedom.
  • #1
83
0
In this excerpt from the notes of Sean M. Carrol, he says:

"Einstein’s equations may be thought of as second-order differential equations for the
metric tensor field gμν. There are ten independent equations (since both sides are symmetric
two-index tensors), which seems to be exactly right for the ten unknown functions of the
metric components. However, the Bianchi identity ∇μGμν = 0 represents four constraints on
the functions Rμν, so there are only six truly independent equations in (4.52). In fact this is
appropriate, since if a metric is a solution to Einstein’s equation in one coordinate system
xμ it should also be a solution in any other coordinate system xμ′ . This means that there are
four unphysical degrees of freedom in gμν (represented by the four functions xμ′ (xμ)), and
we should expect that Einstein’s equations only constrain the six coordinate-independent
degrees of freedom."


[itex]R_{\mu\nu}[/itex]-[itex]\frac{1}{2}[/itex]R[itex]g_{\mu\nu}[/itex]=8[itex]\pi[/itex]G[itex]T_{\mu\nu}[/itex] (Eq. 4.52)

I am confused as to what the 6 independent equations represent. I'm getting lost on thinking that these equations represent 3 spatial coordinates and 1 time coordinate but then what do the other 2 components represent (obviously my understanding is flawed). Any clarification would be greatly appreciated, thanks.
 
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  • #2
Airsteve0 said:
I'm getting lost on thinking that these equations represent 3 spatial coordinates and 1 time coordinate but then what do the other 2 components represent (obviously my understanding is flawed). Any clarification would be greatly appreciated, thanks.

The indices mu and nu *each* take on 4 values, so it's 16 equations, not 4, until you start subtracting out the ones that aren't independent.
 
  • #3
Ok, so once you get down to the 6 which are independent, what do these represent physically?
 
  • #4
I've read that these 6 equations "propagate the 3-metric", if this is correct does this mean they describe time evolution of the spatial portions of the metric?
 
  • #5
Somehow related, IIRC the nr of DOF of GR is 2 and can be obtained in Lagrange formulation as 16 components of metric tensor - 6 because of symmetry of the tensor - 4 from the metricity condition ([itex] \nabla_{\mu} g^{\mu\nu} = 0 [/itex]) - 4 from the Noether identities ([itex] \nabla_{\mu} G^{\mu\nu} = 0 [/itex]) = 2, with G the Einstein tensor. In the Hamiltonian formulation, it's 16-6-8 first class constraints = 2.
Going into quantum mechanics, this 2 are two possible eigenvalues of the 3rd component of the helicity operator, namely -2 and +2.
 
  • #6
thank you for the reply but I feel this is slightly more advanced than what I was looking for in my last question; I am more so looking for an understanding of what the 6 equations physically represent.
 
  • #7
Think about it, what do the Newton's equations represent ? It's the same for the EFE, they are PDE's whose solutions are the components of the metric tensor of the 4D spacetime in the canonical basis.
 
  • #8
Airsteve0 said:
thank you for the reply but I feel this is slightly more advanced than what I was looking for in my last question; I am more so looking for an understanding of what the 6 equations physically represent.

When you count degrees of freedom, you don't necessarily get a unique interpretation. For example, I could define a point in space using (x,y,z), or by giving latitude, longitude, and height above sea level.

So the best we can hope for here is at least one story that successfully accounts for why we have a certain number of d.f.

For physical insight, maybe it would be helpful if we could work out the answer to the corresponding question for Maxwell's equations.
 
  • #9
I think I am starting to understand, thank you bcrowell and dextercioby.
 

1. What are the components of Einstein's equations in 4 dimensions?

The components of Einstein's equations in 4 dimensions are the Einstein tensor, the energy-momentum tensor, and the cosmological constant. These components describe the curvature of spacetime and how it is influenced by matter and energy.

2. How do these components relate to each other?

The Einstein tensor is equal to the energy-momentum tensor multiplied by the gravitational constant and divided by the speed of light squared. The cosmological constant is a constant term that is added to the equations to account for the expansion or contraction of the universe.

3. Can you explain the significance of these equations?

Einstein's equations in 4 dimensions are the cornerstone of general relativity, which is a theory of gravity that describes how matter and energy interact with the fabric of spacetime. They have been used to make accurate predictions about the behavior of objects in the universe, such as the bending of light around massive objects and the existence of black holes.

4. How were these equations developed?

Einstein's equations in 4 dimensions were developed by Albert Einstein in 1915 as a result of his theory of general relativity. He used complex mathematical equations to represent the curvature of spacetime and how it is affected by the presence of matter and energy.

5. Are there any limitations to these equations?

While Einstein's equations in 4 dimensions have been incredibly successful in describing gravity and predicting the behavior of objects in the universe, they do have some limitations. They do not account for the effects of quantum mechanics and cannot fully explain the behavior of very small or very massive objects. Additionally, they do not take into account the expansion of the universe, which is described by a separate set of equations known as the Friedmann equations.

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