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dsaun777
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Can you contract any part of the stress energy tensor with the metric? Say if you had four components Tu1 and contracted that with g^u1 would that produce an invariant?
It has to be the full tensor in order to contract?Dale said:No. That is not a valid tensor operation.
Thank you that's what was thinking.Ibix said:You mean, what is ##g_{a1}T^{a1}##, where summation over ##a## is implied? It's the 1,1 component of ##g_{ab}T^{ac}##, and components of tensors are not invariants.
Which by itself would be just the 1,1 component corresponding to pressure?Ibix said:You mean, what is ##g_{a1}T^{a1}##, where summation over ##a## is implied? It's the 1,1 component of ##g_{ab}T^{ac}=T_b{}^c##, and components of tensors are not invariants.
Not strictly. The contraction of your stress-energy tensor twice with the same basis vector is the pressure across a surface perpendicular to that basis vector.dsaun777 said:Which by itself would be just the 1,1 component corresponding to pressure?
A coordinate independent statement would have to be the full contracted tensor ie. The trace?Ibix said:Not strictly. The contraction of your stress-energy tensor twice with the same basis vector is the pressure across a surface perpendicular to that basis vector.
If you are using an orthonormal basis then that contraction is algebraically equal to the 1,1 component of the tensor. But that's not a coordinate-independent statement and it's sloppy to say that "such and such a component is such and such an observable". It's acceptable-but-sloppy if you specify an orthonormal basis, and technically wrong if you don't.
Ibix said:I don't know about other contractions. ##T^a{}_a## and ##g_{ab}T^{ab}## are the only ones I can think of immediately.
D'oh! Of course they are - I'm just lowering an index (##g_{ab}T^{ac}=T_b{}^c##) then contracting the free indices on that. Do you know if your other one is useful for anything? Edit: it kind of looks like a "modulus-squared" of the tensor, if that makes sense.PeterDonis said:Those are the same thing, they're both the trace. The other obvious one is ##T^{ab} T_{ab}##.
Ibix said:Do you know if your other one is useful for anything?
Why don't you transform the components and check explicitly? :)dsaun777 said:Can you contract any part of the stress energy tensor with the metric? Say if you had four components Tu1 and contracted that with g^u1 would that produce an invariant?
It gets tricky when you change the type of matter you wanthaushofer said:Why don't you transform the components and check explicitly? :)
dsaun777 said:It gets tricky when you change the type of matter you want
Too trickyPeterDonis said:The type of matter doesn't matter. You can easily demonstrate using the general tensor transformation laws that, for example, ##g_{a1} T^{a1}## is not an invariant, regardless of the specific forms of ##g## or ##T##.
If you are unable to do this, then I would strongly suggest that you spend some time learning tensor algebra and developing some facility with standard tensor operations. Sean Carroll's online lecture notes on GR have a good introductory treatment of this in the early chapters.
dsaun777 said:Too tricky
PeterDonis said:I will be happy to re-label this one as "B" level
PeterDonis said:be prepared to have a lot of your threads closed very quickly because there is no point in discussion if you can't make use of it
Tensor contraction with metric for stress energy is a mathematical operation that involves multiplying two tensors and then summing over the repeated indices. The metric tensor is used to raise or lower indices in order to maintain the correct dimensions of the tensors. This operation is commonly used in physics and engineering to calculate stress and energy in a given system.
Tensor contraction with metric is important in physics because it allows us to calculate stress and energy in a given system. These quantities are crucial in understanding the behavior of physical systems and can help us make predictions and solve problems in various fields such as mechanics, electromagnetism, and general relativity.
Einstein's field equations, which describe the relationship between the curvature of spacetime and the distribution of matter and energy, involve tensor contraction with metric. The stress-energy tensor, which represents the distribution of matter and energy in spacetime, is contracted with the metric tensor to calculate the curvature of spacetime.
One example of tensor contraction with metric for stress energy is calculating the stress-energy tensor for a fluid, such as water, in a gravitational field. The stress-energy tensor would involve the density and pressure of the fluid, as well as the gravitational potential, and would be contracted with the metric tensor to calculate the curvature of spacetime caused by the fluid's presence.
Tensor contraction with metric for stress energy is based on the assumptions of general relativity and may not accurately describe systems that involve extreme conditions, such as those found in black holes or during the early universe. Additionally, the calculations involved in tensor contraction can be complex and time-consuming, making it difficult to apply in certain situations.