CITE THIS NOTEBOOK: SpTm: package for Differential Geometry and General Relativity calculations by Bowen Ping. Wolfram Community FEB 27 2023.
GITHUB: https://github.com/Jayce-Ping/SpTm.
GITHUB: https://github.com/Jayce-Ping/SpTm.
SpTm is short for Space-Tim, which is the background of General Relativity. This package is for calculations of tensors expression with abstract index notation. You can use this package to do some calculations when you are learning Differential Geometry or General Relativity. Calculate all components of a tensor of a expression runs out of draft paper and makes it easy to make mistake. But this package will make it easier. In this package, you can use a readable form to input tensors and expression in a easy way.
Introduction
Introduction
I programmed a package for calculations with Wolfram Language, which can do some calculations for differential geometry and general relativity. I like to input an expression in a more readable way. I want to input tensors with subscripts and superscripts, which are able to input to the notebook with ctrl+- and ctrl+6. The English document is as following, in which I introduce the installation steps and usage examples.
Install and Import
Install and Import
Method 1
Method 1
Download SpTm.wl from my GitHub - Jayce-Ping/SpTm manually, place it in the same directory of the current Notebook and run one of the following commands：
In[]:=
Get@FileNameJoin[{NotebookDirectory[],"SpTm.wl"}]
<<SpTm`
Method 2 (recommend)
Method 2 (recommend)
With the Internet connected, run the following command：
URLDownload["https://raw.githubusercontent.com/Jayce-Ping/SpTm/main/SpTm.wl",FileNameJoin[{$UserBaseDirectory,"Applications","SpTm.wl"}]];
The command above will download the SpTm.wl into the right directory. After this, any time you want to import SpTm, just run one of the following commands in any Notebook:
In[]:=
<<SpTm`
In[]:=
Needs["SpTm`"]
Get Help and Usage Information
Get Help and Usage Information
Using the following command to get the usage of every function:
In[]:=
?SpTm`*
Out[]=
Basic Usage
Basic Usage
Input a Tensor
Input a Tensor
Use Ctrl+_ to input the subscript，use Ctrl+^ to input superscript. For example, or. You can use to input the subscripts and use ctrl+6 or ctrl+5 to input the superscripts.And you can use the following function to see what SpTm will do to the expression before:
cd
h
ab
cd
h
ab
ctrl+-
In[]:=
InputExplain
cd
h
ab
Out[]=
STensor[h,{a,b},{c,d}]
InputExplain shows the SpTm Explanation of your input. STensor[h, {sub}, {sup}] represents a tensor with abstract indices, named h, with sub indices {sub} and super indices {sup}.
Simplify an Abstract Expression
Simplify an Abstract Expression
Use STCalculate to transform and simplify an abstract expression:
In[]:=
expr=();
d
v
∇
a
g
bc
e
w
In[]:=
STCalculate[expr]
Out[]=
∇
a
In[]:=
%//FullForm
Out[]//FullForm=
Times[Grad[STensor[w,List[],List[e]],a],STensor[g,List[b,c],List[]],STensor[v,List[],List[d]]]
Because ∇ is the derivative operator adapted to the metric g, the result is as above.
The expression above is the form SpTm deal, which is not that readable for human. We can use ShowForm to make it readable.
The expression above is the form SpTm deal, which is not that readable for human. We can use ShowForm to make it readable.
In[]:=
%//ShowForm
Out[]//StandardForm=
∇
a
e
w
g
bc
d
v
Symmetrize and Antisymmetrize
Symmetrize and Antisymmetrize
You can use STSymmetrize/STAntisymmetrize to symmetrize/antisymmetrize a tensor or an expression as follows:
In[]:=
STSymmetrize[expr,{a,b,c}]
Out[]=
1
6
∇
c
∇
b
∇
c
∇
a
∇
b
∇
a
In[]:=
%//ShowForm
Out[]//StandardForm=
1
6
∇
c
g
ab
e
w
d
v
∇
b
g
ac
e
w
d
v
∇
c
g
ba
e
w
d
v
∇
a
g
bc
e
w
d
v
∇
b
g
ca
e
w
d
v
∇
a
g
cb
e
w
d
v
In[]:=
STAntiSymmetrize[expr,{a,e,d}]
Out[]=
1
6
∇
e
∇
d
∇
e
∇
a
∇
d
∇
a
In[]:=
%//ShowForm
Out[]//StandardForm=
1
6
∇
e
g
bc
d
w
a
v
∇
d
g
bc
e
w
a
v
∇
e
g
bc
a
w
d
v
∇
a
g
bc
e
w
d
v
∇
d
g
bc
a
w
e
v
∇
a
g
bc
d
w
e
v
Set Coordinate System
Set Coordinate System
In fact, most of our work will be done with coordinate system. You can use SetCoordinates to choose a coordinate system:
In[]:=
SetCoordinates[{r,θ,ϕ}]
If there is no warning, the coordinate system has been set.
Coordinate Basis and Dual Coordinate Basis
Coordinate Basis and Dual Coordinate Basis
In Differential Geometry, there is one set of important vectors - coordinate basis, which we usually denote as , where a is the abstract index and is the i-th coordinate basis. But in Mathematica, directly use ∂ will cause a warning and abort like below:
a
∂
∂
i
x
i
x
In[]:=
∂
Besides, it’s complicated for input a vector using . You need to press Ctrl and switch cursor again and again. But there is a meaningless symbol in Mathematica or DifferentialD[] which is able to denote the dual coordinate basis like x or DifferentialD[x] and not encounter a warning.According to this, I find another symbol to denote the coordinate basis in a simple and easy-input way - using or CapitalDifferentialD[].I’ll restate how SpTm denote coordinate basis and dual coordinate basis with examples.Assuming that we already set the coordinate system {r, θ, ϕ}, the coordinate basis will be:,and.
∂
∂
i
x
a
(r)
a
(θ)
a
(ϕ)
a
(r)
a
(θ)
a
(ϕ)
The way to input is to press DD. And input is to press dd. The dual coordinate basis will be:,,and
(r)
a
(θ)
a
(ϕ)
a
(r)
a
(θ)
a
(ϕ)
a
Once you set coordinate system, the coordinate basis and dual coordinate basis will be set automatically.
In[]:=
SetCoordinates[{r, θ, ϕ}]
You can use STensorInfo to get information of each tensor.
In[]:=
{STensorInfo[],STensorInfo[],STensorInfo[]}
(r)
a
(θ)
b
a
(ϕ)
Out[]=
=
,=
,=
r
α
1 |
0 |
0 |
θ
β
0 |
1 |
0 |
α
ϕ
0 |
0 |
1 |
The heads of two expressions are different. But we regard x as a vector and a is its index, so the second form is correct. Remember this. This is the same as x.
Set Metric
Set Metric
Then, we can use SetMetric to set the metric field on this manifold:
If you have already set the coordinate system, you can use the command above. Or, you can set metric and coordinate system together as following:
Examples
Examples
Next, I will use some examples with different metrics to introduce more functions:
2-D Euclidean Metric
2-D Euclidean Metric
Set Metric Components and Coordinate System
Set Metric Components and Coordinate System
Set Tensor Components
Set Tensor Components
Coordinates Transformation
Coordinates Transformation
The second argument is the rule list of current coordinates to the target coordinates.
Verify the Commutation of Coordinate Bases
Verify the Commutation of Coordinate Bases
{u,v} is a set of coordinate bases，{w,v} are a set of orthonormal bases
Calculate the norm of every vector, verify the {w,v} are normal vectors.
2-D Cylinder Metric
2-D Cylinder Metric
Transform the Abstract Expression into Specific Expression
Transform the Abstract Expression into Specific Expression
Here we calculate the geodesics on the cylinder, which are circumference and bus bar.
Calculate the Christoffel Symbol in Current Coordinate System
Calculate the Christoffel Symbol in Current Coordinate System
So Riemann Tensor, Ricci Tensor, Ricci Scalar all vanish. The intrinsic curvature is null but the extrinsic curvature exists.
Calculate the Expression of the Line Element
Calculate the Expression of the Line Element
Calculate the Expression of Volume Element with Coordinates Conditions
Calculate the Expression of Volume Element with Coordinates Conditions
Calculate the Christoffel Symbol and Set as a Tensor
Calculate the Christoffel Symbol and Set as a Tensor
For the result shown above, we can use FullForm to see the full structure of the expression:
The result is a Row. We can use [[1]] to get every parts:
The list includes three parts - tensor expression, equal sign and components array of tensor.
In order to show the components in a beautiful way, It is headed by MatrixForm. So we need to use [[1]] one more time to get the components array in the form of list.
In order to show the components in a beautiful way, It is headed by MatrixForm. So we need to use [[1]] one more time to get the components array in the form of list.
This is the components list.
Calculate the Riemann Tensor and Set as a Tensor
Calculate the Riemann Tensor and Set as a Tensor
Verify the Definition of the Riemann Tensor
Verify the Definition of the Riemann Tensor
Here we notice that the order of indices are different, so we need to transpose one of them and then compare:
Now we can see they are the same, which verify the definition of Riemann Tensor.
Calculate the Ricci Tensor and Set as a Tensor
Calculate the Ricci Tensor and Set as a Tensor
Calculate the Ricci Scalar
Calculate the Ricci Scalar
Calculate the Einstein Tensor and Set as a Tensor
Calculate the Einstein Tensor and Set as a Tensor
Show the Components Information of All Tensors named “R”
Show the Components Information of All Tensors named “R”
Get Components Information with the Input of Abstract Expression Directly
Get Components Information with the Input of Abstract Expression Directly
Verify that the Einstein Tensor of all two-dimensional generalized Riemannian spaces is zero
Verify that the Einstein Tensor of all two-dimensional generalized Riemannian spaces is zero
The components of metric g are functions of coordinates.
Simplify the results with the symmetry of metric g.
Solve the Killing Vector Fields
Solve the Killing Vector Fields
The differential equations here are not easy to solve. Here is a roundabout strategy - guessing that the symmetry of the sphere is a rotational symmetry around three axes in the Cartesian coordinate system.
This is changed to a Cartesian coordinate system in three-dimensional Euclidean space where the three Killing vectors of rotational symmetry are written directly and a coordinate transformation is used to obtain the components in the spherical coordinate system, which are then substituted back here for verification.
This is changed to a Cartesian coordinate system in three-dimensional Euclidean space where the three Killing vectors of rotational symmetry are written directly and a coordinate transformation is used to obtain the components in the spherical coordinate system, which are then substituted back here for verification.
The two-dimensional sphere has three symmetries - rotational symmetry around the x-y-z axis in a Cartesian coordinate system with three independent Killing vector fields.
3-D Euclidean Metric
3-D Euclidean Metric
Rotational Symmetries Corresponding to the Three Killing Vector Fields
Rotational Symmetries Corresponding to the Three Killing Vector Fields
Here three independent Killing vectors can be written based on the rotational symmetry in each coordinate plan:
Do the coordinates transformation:
At this point the specific component expressions for each Killing vector field in the spherical coordinate system are obtained, written back to the corresponding content in the previous spherical metric, and then verified.
Minkowski Spacetime
Minkowski Spacetime
Set the Components, Coordinate System and Symbol of Metric.
Set the Components, Coordinate System and Symbol of Metric.
Verify the Symmetries of Minkowski Spacetime
Verify the Symmetries of Minkowski Spacetime
Verify the Invariance of Space-time Translations
Verify the Invariance of Space-time Translations
Verify Spatial Rotation
Verify Spatial Rotation
Spatial rotation is naturally established. Because the three-dimensional Euclidean space is a submanifold of the Minkowski spacetime, its metric is the 3-D Euclidean metric, which naturally carries three spatial rotation invariants.
Verify Boost Invariance
Verify Boost Invariance
The boost of Minkowski spacetime corresponds to the Lorentz transformation between coordinate system.
Here we verify the t-x boost:
Here we verify the t-x boost:
Besides, you can use BoostMatrix to generate a boost matrix corresponding to a velocity vector like this:
Here I choose geometrized unit system, which means the speed of light is chosen as 1. So the norm of velocity should be less than 1.
We can build the coordinates transformation as following:
We can build the coordinates transformation as following:
We can see the metric didn’t change after boost.
Schwarzschild Metric
Schwarzschild Metric
Calculate the Line Element Expression
Calculate the Line Element Expression
Obviously, when the mass of star M is equal to zero, the spacetime is Minkowski spacetime, which is flat.
Calculation the Volume Element
Calculation the Volume Element
We can use STSpecify to calculate every components of the VolumeElement:
Use TensorSymmetry to check the symmetry of the volume element like:
Calculate Riemann Tensor, Ricci Tensor, Ricci Scalar and Einstein Tensor
Verify the Schwarzschild metric is a solution of vacuum Einstein Field Equation
Calculate Riemann Tensor, Ricci Tensor, Ricci Scalar and Einstein Tensor
Verify the Schwarzschild metric is a solution of vacuum Einstein Field Equation
Verify the Schwarzschild metric is a solution of vacuum Einstein Field Equation
Then, calculate the Weyl Tensor with Schwarzschild metric:
The specific indices here are Greek letters δβαγ corresponding to dbac. So we need to transpose the result to get the indices abcd, and compare it with the result above. The result shows they are equal.
Isotropic Observer - Robertson-Walker Metric
Isotropic Observer - Robertson-Walker Metric
Here I find the Weyl Tensor of Robertson-Walker Metric vanishes, but I don’t know it’s physical meaning yet. If you know, please acknowledge me.
The package may cause bugs and the document is not that good. I will make them better when I’m not busy.
If you are interested in my package and want to know more, please move to https://github.com/Jayce-Ping/SpTm
If you like my package, please star my repository.
If you find any bug, you can open an issue on my GitHub.
If you are interested in my package and want to know more, please move to https://github.com/Jayce-Ping/SpTm
If you like my package, please star my repository.
If you find any bug, you can open an issue on my GitHub.