Example 1. "DiracWeyl"
First create a vector bundle N with base coordinates [t, x, y, z] and fiber coordinates [z1, z2, w1, w2].
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| (2.1) |
Define a metric of signature [+1, -1, -1, -1] and an orthonormal tetrad.
N >
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| (2.2) |
N >
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| (2.3) |
Calculate the solder form.
N >
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![_DG([["tensor", N, [["cov_bas", "con_vrt", "con_vrt"], []]], [[[1, 5, 7], (1/2)*x^2*2^(1/2)], [[1, 6, 8], (1/2)*x^2*2^(1/2)], [[2, 5, 8], (1/2)*2^(1/2)], [[2, 6, 7], (1/2)*2^(1/2)], [[3, 5, 8], -((1/2)*I)*2^(1/2)], [[3, 6, 7], ((1/2)*I)*2^(1/2)], [[4, 5, 7], (1/2)*2^(1/2)], [[4, 6, 8], -(1/2)*2^(1/2)]]])](/support/helpjp/helpview.aspx?si=5668/file05868/math225.png)
| (2.4) |
Define a rank 1-spinor field psi1 and its complex conjugate.
N >
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| (2.5) |
N >
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| (2.6) |
Calculate the Dirac-Weyl energy momentum tensor T.
N >
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![_DG([["tensor", N, [["con_bas", "con_bas"], []]], [[[1, 3], -2^(1/2)*(f(x)^2-h(x)^2)/x^3], [[2, 3], 2^(1/2)*(-h(x)*(diff(f(x), x))+f(x)*(diff(h(x), x)))], [[3, 1], -2^(1/2)*(f(x)^2-h(x)^2)/x^3], [[3, 2], 2^(1/2)*(-h(x)*(diff(f(x), x))+f(x)*(diff(h(x), x)))]]])](/support/helpjp/helpview.aspx?si=5668/file05868/math258.png)
| (2.7) |
Evaluate the Dirac-Weyl field equations E for the given spinor field psi.
N >
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![_DG([["tensor", N, [["con_vrt"], []]], [[[7], ((1/2)*I)*2^(1/2)*((diff(f(x), x))*x+f(x))/x], [[8], -((1/2)*I)*2^(1/2)*((diff(h(x), x))*x+h(x))/x]]]), _DG([["tensor", N, [["con_vrt"], []]], [[[5], -((1/2)*I)*2^(1/2)*((diff(f(x), x))*x+f(x))/x], [[6], ((1/2)*I)*2^(1/2)*((diff(h(x), x))*x+h(x))/x]]])](/support/helpjp/helpview.aspx?si=5668/file05868/math273.png)
| (2.8) |
Check the divergence identity for the dust energy momentum tensor T. The LHS is the covariant divergence of the energy momentum tensor and the RHS is a combination of the field equations.
N >
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![_DG([["tensor", N, [["con_bas"], []]], [[[3], 2^(1/2)*(-x*h(x)*(diff(diff(f(x), x), x))+x*f(x)*(diff(diff(h(x), x), x))+2*f(x)*(diff(h(x), x))-2*h(x)*(diff(f(x), x)))/x]]]), _DG([["tensor", N, [["con_bas"], []]], [[[3], 2^(1/2)*(-x*h(x)*(diff(diff(f(x), x), x))+x*f(x)*(diff(diff(h(x), x), x))+2*f(x)*(diff(h(x), x))-2*h(x)*(diff(f(x), x)))/x]]])](/support/helpjp/helpview.aspx?si=5668/file05868/math290.png)
| (2.9) |
N >
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| (2.10) |
We note that f(x) = h(x) = 1/x is a solution of the Dirac-Weyl field equations:
N >
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| (2.11) |
The covariant divergence of the energy momentum tensor vanishes on this solution:
N >
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| (2.12) |
Example 2. "Dust"
First create a manifold M with base coordinates [t, x, y, z]:
N >
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| (2.13) |
Define a metric.
M >
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| (2.14) |
Define the normalized 4-vector representing the 4-velocity of the dust.
M >
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| (2.15) |
M >
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| (2.16) |
Define the energy density.
M >
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| (2.17) |
Calculate the dust energy- momentum tensor T2.
M >
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![_DG([["tensor", M, [["con_bas", "con_bas"], []]], [[[1, 1], h(t)*cosh(f(t))^2], [[1, 2], -h(t)*cosh(f(t))*sinh(f(t))/t], [[2, 1], -h(t)*cosh(f(t))*sinh(f(t))/t], [[2, 2], h(t)*(cosh(f(t))^2-1)/t^2]]])](/support/helpjp/helpview.aspx?si=5668/file05868/math393.png)
| (2.18) |
Evaluate the dust field equations E2 for the given u and mu.
M >
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![(h(t)*cosh(f(t))+(diff(h(t), t))*cosh(f(t))*t+h(t)*sinh(f(t))*(diff(f(t), t))*t)/t, _DG([["tensor", M, [["con_bas"], []]], [[[1], (cosh(f(t))^2-1+cosh(f(t))*sinh(f(t))*(diff(f(t), t))*t)/t], [[2], -cosh(f(t))*(sinh(f(t))+cosh(f(t))*(diff(f(t), t))*t)/t^2]]])](/support/helpjp/helpview.aspx?si=5668/file05868/math410.png)
| (2.19) |
Check that the follow values for f(t) and h(t) solve the dust field equations.
M >
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| (2.20) |
M >
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| (2.21) |
Check the divergence identity for the dust energy-momentum tensor T2. The LHS is the covariant divergence of the energy-momentum tensor and the RHS is a combination of the field equations.
M >
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| (2.22) |
M >
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| (2.23) |
Example 3. "Electromagnetic"
First create a manifold M with base coordinates [t, x, y, z, t].
M >
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| (2.24) |
Define a metric.
M >
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| (2.25) |
Define an electromagnetic 4-potential A3.
M >
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| (2.26) |
Calculate the electromagnetic energy-momentum tensor T3.
M >
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![_DG([["tensor", M, [["con_bas", "con_bas"], []]], [[[1, 1], -(1/2)*((diff(f2(x), x))^2*x^2+(diff(f1(x), x))^2)/x^4], [[1, 3], (diff(f1(x), x))*(diff(f2(x), x))/x^2], [[2, 2], (1/2)*(-(diff(f2(x), x))^2*x^2+(diff(f1(x), x))^2)/x^2], [[3, 1], (diff(f1(x), x))*(diff(f2(x), x))/x^2], [[3, 3], -(1/2)*((diff(f2(x), x))^2*x^2+(diff(f1(x), x))^2)/x^2], [[4, 4], -(1/2)*(-(diff(f2(x), x))^2*x^2+(diff(f1(x), x))^2)/x^2]]])](/support/helpjp/helpview.aspx?si=5668/file05868/math514.png)
| (2.27) |
Note that the energy-momentum tensor can also be computed from the field strength tensor F = dA.
M >
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| (2.28) |
M >
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![_DG([["tensor", M, [["con_bas", "con_bas"], []]], [[[1, 1], -(1/2)*((diff(f2(x), x))^2*x^2+(diff(f1(x), x))^2)/x^4], [[1, 3], (diff(f1(x), x))*(diff(f2(x), x))/x^2], [[2, 2], (1/2)*(-(diff(f2(x), x))^2*x^2+(diff(f1(x), x))^2)/x^2], [[3, 1], (diff(f1(x), x))*(diff(f2(x), x))/x^2], [[3, 3], -(1/2)*((diff(f2(x), x))^2*x^2+(diff(f1(x), x))^2)/x^2], [[4, 4], -(1/2)*(-(diff(f2(x), x))^2*x^2+(diff(f1(x), x))^2)/x^2]]])](/support/helpjp/helpview.aspx?si=5668/file05868/math534.png)
| (2.29) |
Evaluate the electromagnetic field equations E3 for the given 4-potential A.
M >
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| (2.30) |
Note that the electromagnetic field equations E3 can also be computed from the field strength tensor F = dA.
M >
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| (2.31) |
Check the divergence identity for the electromagnetic energy-momentum tensor T3. The LHS is the covariant divergence of the energy momentum tensor and the RHS is a combination of the matter field equations.
M >
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![_DG([["tensor", M, [["con_bas"], []]], [[[2], (-(diff(f2(x), x))*x^3*(diff(diff(f2(x), x), x))+x*(diff(f1(x), x))*(diff(diff(f1(x), x), x))-(diff(f1(x), x))^2-(diff(f2(x), x))^2*x^2)/x^3]]]), _DG([["tensor", M, [["con_bas"], []]], [[[2], (-(diff(f2(x), x))*x^3*(diff(diff(f2(x), x), x))+x*(diff(f1(x), x))*(diff(diff(f1(x), x), x))-(diff(f1(x), x))^2-(diff(f2(x), x))^2*x^2)/x^3]]])](/support/helpjp/helpview.aspx?si=5668/file05868/math581.png)
| (2.32) |
M >
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| (2.33) |
We note that f1(x) = x^2, f2(x) = ln(x) is a solution of the electromagnetic field equations:
M >
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| (2.34) |
The covariant divergence of the energy-momentum tensor vanishes on this solution:
M >
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| (2.35) |
Example 4. "PerfectFluid"
First create a manifold M with base coordinates [t, x, y, z]:
M >
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| (2.36) |
Define a metric.
M >
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| (2.37) |
Define the normalized 4-velocity.
M >
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| (2.38) |
M >
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| (2.39) |
Define the energy density.
M >
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| (2.40) |
Define the pressure.
M >
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| (2.41) |
Calculate the perfect fluid energy-momentum tensor T4.
M >
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![_DG([["tensor", M, [["con_bas", "con_bas"], []]], [[[1, 1], 5*h(t)+4*k(t)], [[1, 2], 2*(k(t)+h(t))*3^(1/2)/t], [[2, 1], 2*(k(t)+h(t))*3^(1/2)/t], [[2, 2], (2*h(t)+3*k(t))/t^2], [[3, 3], -h(t)], [[4, 4], -h(t)]]])](/support/helpjp/helpview.aspx?si=5668/file05868/math697.png)
| (2.42) |
Evaluate the fluid field equations E4 for the given fluid.
M >
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![_DG([["tensor", M, [["con_bas"], []]], [[[1], (7*k(t)+7*h(t)+5*t*(diff(h(t), t))+4*t*(diff(k(t), t)))/t], [[2], 2*3^(1/2)*(2*k(t)+2*h(t)+t*(diff(k(t), t))+t*(diff(h(t), t)))/t^2]]])](/support/helpjp/helpview.aspx?si=5668/file05868/math710.png)
| (2.43) |
We can use the dsolve command to find the energy density k(t) and the pressure h(t) which satisfy the matter field equations.
M >
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| (2.44) |
M >
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| (2.45) |
Example 5. "Scalar"
First create a manifold M with base coordinates [t, x, y, z].
M >
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| (2.46) |
Define a metric.
M >
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| (2.47) |
Define a scalar field.
M >
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| (2.48) |
Calculate the energy- momentum tensor T5 for the scalar field phi5.
M >
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![_DG([["tensor", M, [["con_bas", "con_bas"], []]], [[[1, 1], (1/2)*(diff(f(t), t))^2-(1/2)*_m^2*f(t)^2], [[2, 2], (1/2)*((diff(f(t), t))^2+_m^2*f(t)^2)/t^2], [[3, 3], (1/2)*(diff(f(t), t))^2+(1/2)*_m^2*f(t)^2], [[4, 4], (1/2)*(diff(f(t), t))^2+(1/2)*_m^2*f(t)^2]]])](/support/helpjp/helpview.aspx?si=5668/file05868/math793.png)
| (2.49) |
Evaluate the matter field equations E for the given scalar field phi.
M >
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| (2.50) |
Check the divergence identity for the scalar energy-momentum tensor T. The LHS is the covariant divergence of the energy-momentum tensor and the RHS is a combination of the matter field equations.
M >
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![_DG([["tensor", M, [["con_bas"], []]], [[[1], (diff(f(t), t))*(diff(f(t), t)+(diff(diff(f(t), t), t))*t-t*_m^2*f(t))/t]]]), _DG([["tensor", M, [["con_bas"], []]], [[[1], (diff(f(t), t))*(diff(f(t), t)+(diff(diff(f(t), t), t))*t-t*_m^2*f(t))/t]]])](/support/helpjp/helpview.aspx?si=5668/file05868/math825.png)
| (2.51) |
M >
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| (2.52) |
Example 6. "NMCScalar"
First create a manifold M with base coordinates [t, x, y, z].
M >
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| (2.53) |
Define a metric.
M >
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| (2.54) |
Define a scalar field
M >
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| (2.55) |
Calculate the energy-momentum tensor T for the non-minimally coupled scalar field phi6.
M >
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| (2.56) |
Evaluate the matter field equations E for the given scalar field phi6.
M >
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| (2.57) |
Check the divergence identity for the scalar energy-momentum tensor T6. The LHS is the covariant divergence of the energy-momentum tensor and the RHS is a combination of the matter field equations.
M >
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![_DG([["tensor", M, [["con_bas"], []]], [[[1], (diff(f(t), t))*(diff(f(t), t)+(diff(diff(f(t), t), t))*t-t*_m^2*f(t))/t]]]), _DG([["tensor", M, [["con_bas"], []]], [[[1], (diff(f(t), t))*(diff(f(t), t)+(diff(diff(f(t), t), t))*t-t*_m^2*f(t))/t]]])](/support/helpjp/helpview.aspx?si=5668/file05868/math922.png)
| (2.58) |
M >
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| (2.59) |