DiffGeo

A Mathematica package to handle everything in differential geometry. From calculating curvature invariants to Hodge star operations on forms, this package can do it all! (That's the aim at least)

So far, the package can be organised into two distinct parts: - Standard GR-invariants - Exterior Algebra

Future Additions

• [x] Conformal Killing Vector
• [x] Spin connection
• [ ] Fix optional arguments when specifying part of the inputs. eg Lie[vec, g, xx, "Down" -> {1, 2}] works but Lie[vec, g, "Down" -> {1, 2}] confuses the option for xx
• [ ] Fix https://github.com/benterre/DiffGeo/issues/1 AntiSelfDualQ returns False on array input
• [x] Allow Form2Tensor to convert tensor-valued n-forms
• [ ] Lie Bracket
• [x] Codifferential of forms
• [x] Laplace-Beltrami Operator
• [ ] Vielbeins
• [ ] Allow Form2Tensor to convert directly to non-coordinate basis
• [ ] Gamma matrices
• [ ] Killing Spinors
• [ ] Fefferman-Graham Gauge

Standard GR-invariants

[!NOTE] For these functions, the metric should be given as a $d\times d$ array and the list of coordinates as a $d$ array.

• Christoffel symbols
• Riemann Tensor
• Ricci Tensor
• Ricci Scalar
• Kretschmann Scalar
• Weyl Tensor
• Euler Scalar
• Cotton Tensor
• Schouten Tensor
• Lovelock Invariants

Christoffel[g, xx]

Calculates the Christoffel symbols for the associated metric g and coordinates xx. Conventions are such that its components are given by spacetime indices $\Gamma^\mu{}_{\nu\rho}$ in that same order. In other words,

Christoffel[g, xx][[\[Mu],\[Nu],\[Rho]]]=$\Gamma^{\mu}{}_{\nu\rho}$.

These are computed from the metric using

$$\Gamma^\mu{\nu\rho}=\frac{1}{2}g^{\mu\sigma}(\partial\nu g{\rho\sigma}+\partial\rho g{\nu\sigma}-\partial\sigma g_{\nu\rho})$$

RiemannTensor[g, xx]

Calculates the Riemann tensor for the associated metric g and coordinates xx. Conventions are such that all spacetime indices are down $R_{\mu\nu\rho\sigma}$. The components are computed from the metric using

math R_{\mu\nu\rho\sigma} = g_{\mu\alpha} \partial_\rho\Gamma^\alpha{}_{\sigma\nu} - \partial_\sigma \Gamma^\alpha{}_{\rho\nu} + \Gamma^{\alpha}{}_{\rho\lambda}\Gamma^\lambda{}_{\sigma\nu} - \Gamma^{\alpha}{}_{\sigma\lambda}\Gamma^{\lambda}{}_{\rho\nu}

RicciTensor[g, xx]

Calculates the Ricci tensor for the associated metric g and coordinates xx. Conventions are such that all spacetime indices are down $R_{\mu\nu}$.

The components are computed from the metric using

$$R{\mu\nu}=g^{\rho\sigma} R{\rho\mu\sigma\nu}$$

RicciScalar[g, xx]

Calculates the Ricci scalar for the associated metric g and coordinates xx.

The components are computed from the metric using

$$R=g^{\mu\nu}R_{\mu\nu}$$

KretschmannScalar[g, xx]

Calculates the Kretschmann scalar for the associated metric g and coordinates xx. Its expression is identical to the third Lovelock invariant at level 2, I32.

The components are computed from the metric using

$$K1 = R{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$$

WeylTensor[g, xx]

Calculates the Weyl tensor for the associated metric g and coordinates xx. Conventions are such that all spacetime indices are down $C_{\mu\nu\rho\sigma}$.

The components are computed from the metric using

$$C{\mu\nu\rho\sigma} = R{\mu\nu\rho\sigma} + \frac{1}{d-2} (R{\mu\sigma}g{\nu\rho} - R{\mu\rho}g{\nu\sigma} + R{\nu\rho}g{\mu\sigma} - R{\nu\sigma}g{\mu\rho}) + \frac{1}{(d-1)(d-2)} R(g{\mu\rho}g{\nu\sigma}- g{\mu\sigma}g{\nu\rho})$$

EulerScalar[g, xx]

Calculates the Euler scalar for the associated metric g and coordinates xx.

The components are computed from the metric through the Weyl tensor as

$$K3 = -C{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} + 2R_{\mu\nu}R^{\mu\nu} - \frac{2}{3}R^2$$

CottonTensor[g, xx]

Calculates the Cotton tensor for the associated metric g and coordinates xx. The components are computed from the metric using

$$C{\mu\nu\rho} = \nabla\rho R{\mu\nu} - \nabla{\nu}R{\mu\rho} +\frac{1}{2(d-1)}(\nabla{\nu} R g{\mu\rho} - \nabla{\rho} R g_{\mu\nu})$$

SchoutenTensor[g, xx]

Calculates the Schouten tensor for the associated metric g and coordinates xx.

The components are computed from the metric using

$$P{\mu\nu} =\frac{1}{d-2} \left( R{\mu\nu} - \frac{R}{2(d-1)} g_{\mu\nu} \right)$$

Lovelock[k, n][g, xx]

Returns the nth Lovelock invariant at order k for the associated metric g and coordinates xx. For n greater than the number of Lovelock invariant at a given order k, Lovelock[n,k] might return additional invariants built from the Weyl tensor (marked in gray below).

| Order | Command | Expression | | :---: | :--- | :--- | | k=1 | L[1,1] | $L^11=R$ | | k=2 | L[2,1] | $L^21=R^2$ | | | L[2,2] | $L^22=R{\mu\nu}R^{\mu\nu}$ | | | L[2,3] | $L^23=R{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=K1$ | | | L[2,4] | $\color{gray}{C{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}}$ | | k=3 | L[3,1] | $L^31=R^3$ | | | L[3,2] | $L^32=R R{\mu\nu}R^{\mu\nu}=R L^22$ | | | L[3,3] | $L^33=R{\mu\alpha} R^{\nu}{}\mu R^{\alpha\mu}$ | | | L[3,4] | $L^34=R{\mu\alpha} R{\nu\beta} R^{\nu\mu\alpha\beta}$ | | | L[3,5] | $L^35=R R{\mu\nu\alpha\beta} R^{\mu\nu\alpha\beta}=R K1$ | | | L[3,6] | $L^36=R{\nu\alpha} R{\beta\gamma\epsilon}{}^\nu R^{\beta\gamma\epsilon\alpha}$ | | | L[3,7] | $L^37=R{\mu\nu\alpha\beta} R^{\mu\nu}{}{\gamma\epsilon} R^{\alpha\beta\gamma\epsilon}$ | | | L[3,8] | $`L^38=R{\mu\nu\alpha\beta} R^{\mu}{}{\gamma}{}^{\alpha}{}{\epsilon} R^{\nu\gamma\beta\epsilon}$ | | |L[3,9]| $\color{gray}{C{\mu\nu\alpha\beta}R^{\mu\alpha}R^{\nu\beta}}$ | | |L[3,10]| $\color{gray}{C{\mu\nu\alpha\beta}C^{\alpha\beta\rho\nu}R^\mu{}\rho}$ | | |L[3,11]| $\color{gray}{C{\mu\nu\alpha\beta}C^{\alpha\beta\rho\sigma}C{\rho\sigma}{}^{\mu\nu}}`$ |

LovelockAll[g, xx]

Returns an array containing all Lovelock Invariant available in this package.

• LovelockAll[g, xx, "Extend"->True]: includes invariants built from the Weyl tensor (default)
• LovelockAll[g, xx, "Extend"->False]: doesn't include invariants built from the Weyl tensor

GRInvariants[g, xx]

Returns an array containing all the scalar curvature invariant available in this package.

• GRInvariants[g, xx, "Minimal"->True]: only returns a basis of multiplicatively independent invariants
• GRInvariants[g, xx, "Minimal"->False]: returns all invariants (default)

Exterior Algebra

[!NOTE] For this part of the package to work properly, avoid changing the definition of the exterior derivative d as well as the wedge product Wedge

Based on the EDC360 package. The exterior algebra part of this package defines the exterior derivative as d. The Wedge product can be used on forms as d[x]\[Wedge]d[y].

• Form2Metric
• Form2Tensor
• Tensor2Form
• HodgeStar
• cod
• LaplaceBeltrami
• SelfDualQ
• AntiSelfDualQ
• Lie
• KillingQ
• Cov
• ConfKillingQ
• InteriorProduct
• SpinConnection

Form2Metric

Form2Metric[g,xx]: Converts a metric, (abusively) written using squared one-forms, into its matrix representation, using the set of coordinates xx.

Example ``` g = d[r]^2 + r^2 d[[Theta]]^2; xx = {r,[Theta]};

Form2Metric[g, xx] `` {{1,0},{0,r^2}}`

Form2Tensor

Form2Tensor[form, xx]: Converts a d-form form, represented using the exterior derivative d, into its tensor components, using xx as a list of coordinates.

Form2Tensor[form] will try to use a globally defined variable called xx as the set of coordinates and then call Form2Tensor[form, xx].

Optional Arguments: - Assumptions->None: Array of assumptions used in Simplify - "Type"->String. When equal to 'SymmetrizedArray' returns the tensor as a fully anti-symmetric SymmetrizedArray. When set to 'Normal' returns a standard array. (defaults to 'Normal')

Example ``` form = d[x][Wedge]d[y]; xx = {x,y};

Form2Tensor[form, xx] `` {{0,1},{-1,0}}`

Tensor2Form

Tensor2Form[tensor, xx]: Converts a tensor tensor of any rank into its form representation using xx as a list of coordinates.

Tensor2Form[tensor] will try to use a globally defined variable called xx as the set of coordinates and then call Tensor2Form[tensor, xx].

If the original tensor has a symmetric part, it is automatically set to zero.

Optional Arguments: - Assumptions->None: Array of assumptions used in Simplify

Example ``` tensor = {{0,1},{-1,0}}; xx = {w,z};

Tensor2Form[tensor, xx] `` d[w][Wedge]d[z]`

HodgeStar

HodgeStar[tensor, g]: Returns the Hodge dual of the tensor tensor, with respect to the metric g.

HodgeStar[form, g, xx]: Converts the form form to its tensor representation then calls HodgeStar[tensor, metric] using the list of coordinates xx.

If metric isn't specified, HodgeStar will try to use a globally defined variable called g as the metric.

Optional Arguments: - Assumptions->None: Array of assumptions used in Simplify

Example ``` xx = {r,[Theta],[Phi]}; g = Form2Metric[d[r]^2 + r^2 (d[[Theta]]^2 + Sin[[Theta]]^2 d[[Phi]]^2), xx];

form = d[r][Wedge]d[[Theta]];

HodgeStar[form, g, xx] HodgeStar[form, g, xx, Assumptions -> {r > 0, [Theta] > 0, [Theta] < [Pi]}] `` (Sqrt[Abs[r^4 Sin[[Theta]]^2]] d[[Phi]])/r^2`

d[\[Phi]] Sin[\[Theta]]

cod

cod[form, g, xx]: Returns the codifferential of the form, given a metric g and coordinates xx.

If $\omega$ is a $k$-form on a $d$-dimensional manifold with metric signature $s$, then its codifferential is calculated using

math d^\dagger\omega = (-1)^{d(k+1)+1}s\star d\star\omega

Optional Arguments: - Assumptions->None: Array of assumptions used in Simplify

LaplaceBeltrami

LaplaceBeltrami[form, g, xx]: Returns the Laplace-Beltrami operator acted on the form, given a metric g and coordinates xx.

If $\omega$ is a $k$-form on a manifold with metric $g$, then Laplace-Beltrami operator is defined as

math \Delta\omega = (d^\dagger d + d d^\dagger)\omega = (d+d^\dagger)^2\omega

Optional Arguments: - Assumptions->None: Array of assumptions used in Simplify

SelfDualQ

SelfDualQ[form, g, xx]: Return True if form is self-dual wrt the metric g and False otherwise.

Accepts rank-r tensor and r-form representation for the input. The former doesn't require the coordinates xx to be specified.

SelfDualQ[form] will try to use globally defined variables called xx as the set of coordinates and g as the metric and then calls SelfDualQ[form, g, xx].

Optional Arguments: - Assumptions->None: Array of assumptions used in Simplify

Example ``` xx = {x1, x2, x3, x4}; g = IdentityMatrix[4];

form1 = d[x1][Wedge]d[x2] - d[x3][Wedge]d[x4]; form2 = d[x1][Wedge]d[x2] + d[x3][Wedge]d[x4];

SelfDualQ[form1, g, xx] SelfDualQ[form2, g, xx] `` False`

True

AntiSelfDualQ

AntiSelfDualQ[form, g, xx]: Return True if form is anti-self-dual wrt the metric g and False otherwise.

Accepts rank-r tensor and r-form representation for the input. The former doesn't require the coordinates xx to be specified.

AntiSelfDualQ[form] will try to use globally defined variables called xx as the set of coordinates and g as the metric and then calls AntiSelfDualQ[form, g, xx].

Optional Arguments: - Assumptions->None: Array of assumptions used in Simplify

Example ``` xx = {x1, x2, x3, x4}; g = IdentityMatrix[4];

form1 = d[x1][Wedge]d[x2] - d[x3][Wedge]d[x4]; form2 = d[x1][Wedge]d[x2] + d[x3][Wedge]d[x4];

AntiSelfDualQ[form1, g, xx] AntiSelfDualQ[form2, g, xx] `` True`

False

Lie

Lie[vec, tensor, xx]: Returns the Lie derivative of tensor with respect to the vector vec using the set of coordinates xx. A unit vector in the coordinates can be used as vector input by specifying an integer (in the range 1 to Length[xx]) instead of vec. A form of degree r can be used in lieu of tensor, after which it is converted to a tensor of rank r.

Lie[vec, tensor] will try to use a globally defined variable called xx as the set of coordinates and then call Lie[vec, tensor, xx].

Let tensor be a $(p,q)$-tensor, whose components are denoted by $T^{\mu_1\cdots \mu_p}{}_{\nu_1\cdots\nu_q}$, and let's denote the components of vec by $\xi^\mu$. Then Lie[vec, tensor, xx] is also a $(p,q)$-tensor with components

math (\mathcal{L}_{\xi}T)^{\mu_1\cdots \mu_p}{}_{\nu_1\cdots\nu_q} = \xi^\rho\partial_\rho T^{\mu_1\cdots \mu_p}{}_{\nu_1\cdots\nu_q} - \partial_\rho\xi^{\mu_1}T^{\rho\mu_2\cdots \mu_p}{}_{\nu_1\cdots\nu_q} - \cdots - \partial_\rho\xi^{\mu_p}T^{\mu_\cdots \mu_{p-1}\rho}{}_{\nu_1\cdots\nu_q} + \partial_{\nu_1}\xi^{\rho}T^{\mu_1\cdots \mu_p}{}_{\rho\nu_2\cdots\nu_q} + \cdots + \partial_{\nu_q}\xi^{\rho}T^{\mu_1\cdots \mu_p}{}_{\nu_1\cdots\nu_{q-1}\rho}

Optional Arguments: - Assumptions->None: Array of assumptions used in Simplify - "Up"->None: Array containing the set of indices in 'tensor' which are contravariant (or up) - "Down"->Range[rank]: Array containing the set of indices in 'tensor' which are covariant (or down) - Type->"Other": Used to specify the output format. Type->"Form" will output the tensor as a rank-r form (defaults to that when using a form as input)

Example ``` xx = {r, [Theta], [Phi]}; g = DiagonalMatrix[{1, r^2, r^2 Sin[[Theta]]^2}];

vec = {r^2, [Theta], 0}; tensor = {{r, [Theta]}, {Sin[[Theta]], Cos[[Theta]]}};

Lie[vec, tensor] `` {{r^2, [Theta]}, {[Theta] Cos[[Theta]], -[Theta] Sin[[Theta]]}}`

Example ``` xx = {r, [Theta], [Phi]}; g = DiagonalMatrix[{1, r^2, r^2 Sin[[Theta]]^2}];

Lie[1, g] Lie[3, g] `` {{0, 0, 0}, {0, 2 r, 0}, {0, 0, 2 r Sin[[Theta]]^2}}`

{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}

KillingQ

KillingQ[vector, g, xx]: Returns True if the vector is Killing with respect to the metric g given the set of coordinates xx, and False otherwise.

KillingQ[vector, g] will try to use a globally defined variable called xx as the set of coordinates and then calls KillingQ[vector, g, xx].

KillingQ[vector] will try to use a globally defined variable called g as the metric and then calls KillingQ[vector, g].

A vector field $V=V^\mu\partial_\mu$ is Killing if the Lie derivative of the metric along it vanishes, $\mathcal{L}_{V}g=0$.

Optional Arguments: - Assumptions->None: Array of assumptions used in Simplify

Cov

Cov[vec, tensor, g, xx]: Returns the covariant derivative of tensor with respect to the vector vec using the Christoffel symbols built from the metric g and set of coordinates xx. A unit vector in the coordinates can be used as vector input by specifying an integer (in the range 1 to Length[xx]) instead of vec. A form of degree r can be used in lieu of tensor, after which it is converted to a tensor of rank r.

Cov[vec, tensor, g] will try to use a globally defined variable called xx as the set of coordinates and then calls Lie[vec, tensor, g, xx]. Cov[vec, tensor] will try to use a globally defined variable called g as the metric and then calls Lie[vec, tensor, g].

Let tensor be a $(p,q)$-tensor, whose components are denoted by $T^{\mu_1\cdots \mu_p}{}_{\nu_1\cdots\nu_q}$, and let's denote the components of vec by $\xi^\mu$. Then Cov[vec, tensor, xx] is also a $(p,q)$-tensor with components

math (\nabla_{\xi}T)^{\mu_1\cdots \mu_p}{}_{\nu_1\cdots\nu_q} = \xi^\rho\partial_\rho T^{\mu_1\cdots \mu_p}{}_{\nu_1\cdots\nu_q} + \xi^{\rho}\Gamma^{\mu_1}{}_{\rho\sigma}T^{\sigma\mu_2\cdots \mu_p}{}_{\nu_1\cdots\nu_q} + \cdots + \xi^{\rho}\Gamma^{\mu_p}{}_{\rho\sigma}T^{\mu_\cdots \mu_{p-1}\sigma}{}_{\nu_1\cdots\nu_q} - \xi^{\rho}\Gamma^{\sigma}{}_{\rho\nu_1}T^{\mu_1\cdots \mu_p}{}_{\sigma\nu_2\cdots\nu_q} - \cdots - \xi^{\rho}\Gamma^{\sigma}{}_{\rho\nu_q}T^{\mu_1\cdots \mu_p}{}_{\nu_1\cdots\nu_{q-1}\sigma}

Optional Arguments: - Assumptions->None: Array of assumptions used in Simplify - "Up"->None: Array containing the set of indices in 'tensor' which are contravariant (or up) - "Down"->Range[rank]: Array containing the set of indices in 'tensor' which are covariant (or down) - Type->"Other": Used to specify the output format. Type->"Form" will output the tensor as a rank-r form (defaults to that when using a form as input)

Example ```

```

ConfKillingQ

ConfKillingQ[vector, g, xx]: Returns True if the vector is conformal Killing with respect to the metric g given the set of coordinates xx, and False otherwise.

ConfKillingQ[vector, g] will try to use a globally defined variable called xx as the set of coordinates and then calls ConfKillingQ[vector, g, xx].

ConfKillingQ[vector] will try to use a globally defined variable called g as the metric and then calls ConfKillingQ[vector, g].

A vector field $V=V^\mu\partial_\mu$ is conformal Killing if it obeys the equation math \mathcal{L}_V g=\frac{2}{d}\nabla_\mu V^\mu g

Optional Arguments: - Assumptions->None: Array of assumptions used in Simplify

InteriorProduct

InteriorProduct[vec, form, xx]: Returns the interior product of the form with respect to the vector vec given the set of coordinates xx.

InteriorProduct[vec, tensor, xx]: Returns the interior product of the tensor with respect to the vector vec given the set of coordinates xx.

Optional Arguments: - Assumptions->None: Array of assumptions used in Simplify - Type->"Other": Used to specify the ouput format. Type->"Form" will output a rank r-1 form (defaults to that when using rank-r form as input)

SpinConnection

SpinConnection[e, xx]: Returns the spin connection one-form $\omega^{ab}$ with respect to the array of vielbein one-forms e and coordinate list xx. The vielbeins e can also be specified using a matrix with non-coordinate components first, i.e. $e^a{}_\mu$.

Its components are defined via math \omega_\mu{}^{ab}=e^a{}_\nu(\partial_\mu e^{\nu b}+\Gamma^{\nu}{}_{\mu\rho}e^{\rho b})

Optional Arguments: - Assumptions->None: Array of assumptions used in Simplify

Example ``` xx = {[Theta], [CurlyPhi], [Tau], t}; e = {Sin[[CurlyPhi] + [Tau]] d[[Theta]] + Cos[[Theta]] Sin[[Theta]] Cos[[CurlyPhi] + [Tau]] d[[CurlyPhi]] [Minus] Cos[[Theta]] Sin[[Theta]] Cos[[CurlyPhi] + [Tau]] d[[Tau]], -Cos[[CurlyPhi] + [Tau]] d[[Theta]] + Cos[[Theta]] Sin[[Theta]] Sin[[CurlyPhi] + [Tau]] d[[CurlyPhi]] - Cos[[Theta]] Sin[[Theta]] Sin[[CurlyPhi] + [Tau]] d[[Tau]], Sin[[Theta]]^2 d[[CurlyPhi]] + Cos[[Theta]]^2 d[[Tau]], [Beta] d[t]};

SpinConnection[e, xx] `` {{0, Cos[[Theta]]^2 d[[Tau]] + d[[CurlyPhi]] Sin[[Theta]]^2, Cos[[Tau] + [CurlyPhi]] d[[Theta]] + Cos[[Theta]] (d[[Tau]] - d[[CurlyPhi]]) Sin[[Theta]] Sin[[Tau] + [CurlyPhi]], 0}, {-Cos[[Theta]]^2 d[[Tau]] - d[[CurlyPhi]] Sin[[Theta]]^2, 0, -Cos[[Theta]] Cos[[Tau] + [CurlyPhi]] d[[Tau]] Sin[[Theta]] + Cos[[Theta]] Cos[[Tau] + [CurlyPhi]] d[[CurlyPhi]] Sin[[Theta]] + d[[Theta]] Sin[[Tau] + [CurlyPhi]], 0}, {-Cos[[Tau] + [CurlyPhi]] d[[Theta]] + Cos[[Theta]] (-d[[Tau]] + d[[CurlyPhi]]) Sin[[Theta]] Sin[[Tau] + [CurlyPhi]], Cos[[Theta]] Cos[[Tau] + [CurlyPhi]] d[[Tau]] Sin[[Theta]] - 1/2 Cos[[Tau] + [CurlyPhi]] d[[CurlyPhi]] Sin[2 [Theta]] - d[[Theta]] Sin[[Tau] + [CurlyPhi]], 0, 0}, {0, 0, 0, 0}}`