7.6. Product metrics

Spacetimes with a high degree of symmetry can be sometimes locally decomposed as products of reduced manifolds and the orbits of symmetry. The metric of the whole manifold can then be given in terms of metrics on simpler vbundles. Currently there is a stupid limitation to two subvbundles. It will be dropped soon.

Define a product metric. We already have a scalar r[] on M3. Now we define another scalar w[] on S2:

In[612]:=

DefTensor[w[], S2]

** DefTensor: Defining tensor w[] .

In[613]:=

Catch @ DefProductMetric[g5[-μ, -ν], {{TangentM3, w[]}, {TangentS2, r[]}}, Cd5, {"#", ""}]

DefProductMetric :: nometric : VBundle TangentS2 does not have a metric.

In[614]:=

DefMetric[1, gamma[-A, -B], CdS, {":", "D"}, PrintAs→"γ"]

** DefTensor: Defining symmetric metric tensor gamma[-A, -B] .

** DefTensor: Defining antisymmetric tensor epsilongamma[A, B] .

** DefCovD: Defining covariant derivative CdS[-A] .

** DefTensor: Defining vanishing torsion tensor TorsionCdS[A, -B, -C] .

** DefTensor: Defining symmetric Christoffel tensor ChristoffelCdS[A, -B, -C] .

** DefTensor: Defining Riemann tensor RiemannCdS[-A, -B, -C, -D] .

** DefTensor: Defining symmetric Ricci tensor RicciCdS[-A, -B] .

** DefCovD:  Contractions of Riemann automatically replaced by Ricci.

** DefTensor: Defining Ricci scalar RicciScalarCdS[] .

** DefCovD:  Contractions of Ricci automatically replaced by RicciScalar.

** DefTensor: Defining vanishing Einstein tensor EinsteinCdS[-A, -B] .

** DefTensor: Defining vanishing Weyl tensor WeylCdS[-A, -B, -C, -D] .

** DefTensor: Defining vanishing TFRicci tensor TFRicciCdS[-A, -B] .

** DefCovD:  Computing RiemannToWeylRules for dim 2

** DefCovD:  Computing RicciToTFRicci for dim 2

** DefCovD:  Computing RicciToEinsteinRules for dim 2

We define the warped metric      w[]^2 MetricOfVBundle[TangentM3]+r[]^2 MetricOfVBundle[TangentS2] :

In[615]:=

DefProductMetric[g5[-μ, -ν], {{TangentM3, w[]}, {TangentS2, r[]}}, Cd5, {"#", ""}]

** DefTensor: Defining symmetric metric tensor g5[-μ, -ν] .

** DefTensor: Defining antisymmetric tensor epsilong5[η, λ, μ, ν, ρ] .

** DefCovD: Defining covariant derivative Cd5[-μ] .

** DefTensor: Defining vanishing torsion tensor TorsionCd5[η, -λ, -μ] .

** DefTensor: Defining symmetric Christoffel tensor ChristoffelCd5[η, -λ, -μ] .

** DefTensor: Defining Riemann tensor RiemannCd5[-η, -λ, -μ, -ν] .

** DefTensor: Defining symmetric Ricci tensor RicciCd5[-η, -λ] .

** DefTensor: Defining Ricci scalar RicciScalarCd5[] .

** DefTensor: Defining symmetric Einstein tensor EinsteinCd5[-η, -λ] .

** DefTensor: Defining Weyl tensor WeylCd5[-η, -λ, -μ, -ν] .

   Rules {1, 2, 3, 4, 5, 6, 7, 8} have been declared as DownValues for WeylCd5 .

** DefTensor: Defining symmetric TFRicci tensor TFRicciCd5[-η, -λ] .

   Rules {1, 2} have been declared as DownValues for TFRicciCd5 .

** DefCovD:  Computing RiemannToWeylRules for dim 5

** DefCovD:  Computing RicciToTFRicci for dim 5

** DefCovD:  Computing RicciToEinsteinRules for dim 5

ExpandProducMetric        Expansion of the metric of a product manifold into objects of its submanifolds

Computations with product metrics.

The delta tensors are automatically expanded:

In[616]:=

{g5[a, -b], g5[A, -B]}

Out[616]=

{δ_ ( b)^a , δ_ ( B)^A }

In[617]:=

InputForm/@%

Out[617]=

{delta[a, -b], delta[A, -B]}

Now we can compute any object on M5 in terms of objects of M3 and S2:

In[618]:=

ExpandProductMetric[{g5[-a, -b], g5[-a, -B], g5[-A, -B]}, g5]

Out[618]=

{g_ab^   w_^^2, 0, γ_AB^   r_^^2}

In[619]:=

Out[619]=

In[620]:=

RiemannCd5[-a, -b, -c, -d]//ExpandProductMetric//ContractMetric//Simplify

Out[620]=

(w_^^2 (r_^^2 R[▽] _abcd^     + γ_  ^AB (g_ad^   g_bc^   - g_ac^   g_bd^  ) (D_A^ w_^) (D_B^ w_^)))/r_^^2

In[621]:=

RicciScalarCd5[]//ExpandProductMetric//ContractMetric//Simplification

Out[621]=

By default derivative indices cannot be contracted with metric tensors. This behaviour can be changed using the option AllowUpperDerivatives:

In[622]:=

ContractMetric[%, AllowUpperDerivatives→True]//Simplification

Out[622]=

(r_^^2 R[▽] _^ + R[D] _^ w_^^2 - 4 r_^ (▽_a^ ▽_ ^ar_^) - 2 (▽_a^ r_^) (▽_ ^ar_^) - 6 w_^ (D_A^ D_ ^Aw_^) - 6 (D_A^ w_^) (D_ ^Aw_^))/(r_^^2 w_^^2)

Clean up

In[623]:=

UndefMetric[g5]

** UndefTensor: Undefined antisymmetric tensor epsilong5

** UndefTensor: Undefined symmetric Christoffel tensor ChristoffelCd5

** UndefTensor: Undefined symmetric Einstein tensor EinsteinCd5

** UndefTensor: Undefined symmetric Ricci tensor RicciCd5

** UndefTensor: Undefined Ricci scalar RicciScalarCd5

** UndefTensor: Undefined Riemann tensor RiemannCd5

** UndefTensor: Undefined symmetric TFRicci tensor TFRicciCd5

** UndefTensor: Undefined vanishing torsion tensor TorsionCd5

** UndefTensor: Undefined Weyl tensor WeylCd5

** UndefCovD: Undefined covariant derivative Cd5

** UndefTensor: Undefined symmetric metric tensor g5

In[624]:=

UndefMetric[metricg]

** UndefTensor: Undefined antisymmetric tensor epsilonmetricg

** UndefTensor: Undefined symmetric Christoffel tensor ChristoffelCD

** UndefTensor: Undefined symmetric Einstein tensor EinsteinCD

** UndefTensor: Undefined symmetric Ricci tensor RicciCD

** UndefTensor: Undefined Ricci scalar RicciScalarCD

** UndefTensor: Undefined Riemann tensor RiemannCD

** UndefTensor: Undefined symmetric TFRicci tensor TFRicciCD

** UndefTensor: Undefined vanishing torsion tensor TorsionCD

** UndefTensor: Undefined vanishing Weyl tensor WeylCD

** UndefCovD: Undefined covariant derivative CD

** UndefTensor: Undefined symmetric metric tensor metricg

In[625]:=

UndefMetric[gamma]

** UndefTensor: Undefined antisymmetric tensor epsilongamma

** UndefTensor: Undefined symmetric Christoffel tensor ChristoffelCdS

** UndefTensor: Undefined vanishing Einstein tensor EinsteinCdS

** UndefTensor: Undefined symmetric Ricci tensor RicciCdS

** UndefTensor: Undefined Ricci scalar RicciScalarCdS

** UndefTensor: Undefined Riemann tensor RiemannCdS

** UndefTensor: Undefined vanishing TFRicci tensor TFRicciCdS

** UndefTensor: Undefined vanishing torsion tensor TorsionCdS

** UndefTensor: Undefined vanishing Weyl tensor WeylCdS

** UndefCovD: Undefined covariant derivative CdS

** UndefTensor: Undefined symmetric metric tensor gamma

In[626]:=

UndefManifold[M5]

** UndefVBundle: Undefined vbundle TangentM5

** UndefManifold: Undefined manifold M5

Created by Mathematica  (May 16, 2008) Valid XHTML 1.1!