trace of einstein equation - general relativity
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I know quite well what the trace of a matrix is; however, I am not quite sure I understand the meaning of the 'trace' concept when applied to tensors. I would be very grateful to you if:
How can i prove that the trace of einstein equation in general relativity is zero? And how can i find the values of the elements of the principal diagonal? There have 4 elements in the diagonal, thats right ?
general-relativity
asked Apr 16 '16 at 14:39
Lucas G Leite F PollitoLucas G Leite F Pollito
12
$endgroup$
add a comment |
$begingroup$
I know quite well what the trace of a matrix is; however, I am not quite sure I understand the meaning of the 'trace' concept when applied to tensors. I would be very grateful to you if:
How can i prove that the trace of einstein equation in general relativity is zero? And how can i find the values of the elements of the principal diagonal? There have 4 elements in the diagonal, thats right ?
general-relativity
asked Apr 16 '16 at 14:39
Lucas G Leite F PollitoLucas G Leite F Pollito
12
$endgroup$
1
$begingroup$
"How can I prove that the trace of Einstein equation in general relativity is zero?" you cant because in general it is not.
$endgroup$
– AccidentalFourierTransform
Apr 16 '16 at 14:53
$begingroup$
if i have only two dimension ? can you write the components of the matrix ??
$endgroup$
– Lucas G Leite F Pollito
Apr 18 '16 at 1:11
add a comment |
$begingroup$
I know quite well what the trace of a matrix is; however, I am not quite sure I understand the meaning of the 'trace' concept when applied to tensors. I would be very grateful to you if:
How can i prove that the trace of einstein equation in general relativity is zero? And how can i find the values of the elements of the principal diagonal? There have 4 elements in the diagonal, thats right ?
general-relativity
asked Apr 16 '16 at 14:39
Lucas G Leite F PollitoLucas G Leite F Pollito
12
$endgroup$
I know quite well what the trace of a matrix is; however, I am not quite sure I understand the meaning of the 'trace' concept when applied to tensors. I would be very grateful to you if:
How can i prove that the trace of einstein equation in general relativity is zero? And how can i find the values of the elements of the principal diagonal? There have 4 elements in the diagonal, thats right ?
general-relativity
general-relativity
asked Apr 16 '16 at 14:39
Lucas G Leite F PollitoLucas G Leite F Pollito
12
asked Apr 16 '16 at 14:39
Lucas G Leite F PollitoLucas G Leite F Pollito
12
asked Apr 16 '16 at 14:39
Lucas G Leite F PollitoLucas G Leite F Pollito
12
asked Apr 16 '16 at 14:39
Lucas G Leite F PollitoLucas G Leite F Pollito
12
asked Apr 16 '16 at 14:39
Lucas G Leite F PollitoLucas G Leite F Pollito
12
12
1
$begingroup$
"How can I prove that the trace of Einstein equation in general relativity is zero?" you cant because in general it is not.
$endgroup$
– AccidentalFourierTransform
Apr 16 '16 at 14:53
$begingroup$
if i have only two dimension ? can you write the components of the matrix ??
$endgroup$
– Lucas G Leite F Pollito
Apr 18 '16 at 1:11
add a comment |
1
$begingroup$
"How can I prove that the trace of Einstein equation in general relativity is zero?" you cant because in general it is not.
$endgroup$
– AccidentalFourierTransform
Apr 16 '16 at 14:53
$begingroup$
if i have only two dimension ? can you write the components of the matrix ??
$endgroup$
– Lucas G Leite F Pollito
Apr 18 '16 at 1:11
1
1
$begingroup$
"How can I prove that the trace of Einstein equation in general relativity is zero?" you cant because in general it is not.
$endgroup$
– AccidentalFourierTransform
Apr 16 '16 at 14:53
$begingroup$
"How can I prove that the trace of Einstein equation in general relativity is zero?" you cant because in general it is not.
$endgroup$
– AccidentalFourierTransform
Apr 16 '16 at 14:53
$begingroup$
if i have only two dimension ? can you write the components of the matrix ??
$endgroup$
– Lucas G Leite F Pollito
Apr 18 '16 at 1:11
$begingroup$
if i have only two dimension ? can you write the components of the matrix ??
$endgroup$
– Lucas G Leite F Pollito
Apr 18 '16 at 1:11
add a comment |
1 Answer
1
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$begingroup$
For a (0,2)-tensor $T_{ab}$ say, its trace is $T = g^{ab}T_{ab}$ which uses the inverse metric components. If instead you write this tensor as a (1,1)-tensor $T_a^b$, then its trace is simply the usual trace of a matrix: $T^a_a$.
When you mention the Einstein field equations, do you mean instead local conservation of energy: $nabla_a T^{ab}=0$? See e.g. Wikipedia
answered Jan 6 at 8:09
Colin MacLaurinColin MacLaurin
1065
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add a comment |
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$begingroup$
For a (0,2)-tensor $T_{ab}$ say, its trace is $T = g^{ab}T_{ab}$ which uses the inverse metric components. If instead you write this tensor as a (1,1)-tensor $T_a^b$, then its trace is simply the usual trace of a matrix: $T^a_a$.
When you mention the Einstein field equations, do you mean instead local conservation of energy: $nabla_a T^{ab}=0$? See e.g. Wikipedia
answered Jan 6 at 8:09
Colin MacLaurinColin MacLaurin
1065
$endgroup$
add a comment |
$begingroup$
For a (0,2)-tensor $T_{ab}$ say, its trace is $T = g^{ab}T_{ab}$ which uses the inverse metric components. If instead you write this tensor as a (1,1)-tensor $T_a^b$, then its trace is simply the usual trace of a matrix: $T^a_a$.
When you mention the Einstein field equations, do you mean instead local conservation of energy: $nabla_a T^{ab}=0$? See e.g. Wikipedia
answered Jan 6 at 8:09
Colin MacLaurinColin MacLaurin
1065
$endgroup$
add a comment |
$begingroup$
For a (0,2)-tensor $T_{ab}$ say, its trace is $T = g^{ab}T_{ab}$ which uses the inverse metric components. If instead you write this tensor as a (1,1)-tensor $T_a^b$, then its trace is simply the usual trace of a matrix: $T^a_a$.
When you mention the Einstein field equations, do you mean instead local conservation of energy: $nabla_a T^{ab}=0$? See e.g. Wikipedia
answered Jan 6 at 8:09
Colin MacLaurinColin MacLaurin
1065
$endgroup$
For a (0,2)-tensor $T_{ab}$ say, its trace is $T = g^{ab}T_{ab}$ which uses the inverse metric components. If instead you write this tensor as a (1,1)-tensor $T_a^b$, then its trace is simply the usual trace of a matrix: $T^a_a$.
When you mention the Einstein field equations, do you mean instead local conservation of energy: $nabla_a T^{ab}=0$? See e.g. Wikipedia
answered Jan 6 at 8:09
Colin MacLaurinColin MacLaurin
1065
answered Jan 6 at 8:09
Colin MacLaurinColin MacLaurin
1065
answered Jan 6 at 8:09
Colin MacLaurinColin MacLaurin
1065
answered Jan 6 at 8:09
Colin MacLaurinColin MacLaurin
1065
1065
add a comment |
add a comment |
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"How can I prove that the trace of Einstein equation in general relativity is zero?" you cant because in general it is not.
$endgroup$
– AccidentalFourierTransform
Apr 16 '16 at 14:53
$begingroup$
if i have only two dimension ? can you write the components of the matrix ??
$endgroup$
– Lucas G Leite F Pollito
Apr 18 '16 at 1:11