Cadabra Q&A - Recent questions

declare a mapping with previous expression

Fri, 23 Jan 2026 16:12:30 +0000

I have some expression reached after a long list of manipulations with some free indices, and I want to create a mapping (I think documentation prefers calling them "substitution rules") like:

M_{\mu} -> <previous expression with only \mu as free indice>

I tried doing:

M_mapping := M_{\mu} -> $(_);

but this gives:
`\begin{verbatim}RuntimeError: Python object '' does not exist.

At: Notebook Cell (Line 1): Mmapping = Ex(' M{\mu} -> @(_)') \end{verbatim}`

I tried doing:

M_expr = _;   <--- this succeeds
M_mapping := M_{\mu} -> @(M_expr); <--- this gives an error:

`\begin{verbatim}RuntimeError: Python object '\prod' does not exist.

At: Notebook Cell (Line 1): Mmapping = Ex(' M{\mu} -> @(M_expr)') \end{verbatim}`

What's the correct syntax to create a mapping with the previous expression?

canonicalise gets stuck on swapped indices

Wed, 21 Jan 2026 19:22:39 +0000

The following snippet shows the issue I'm having right now:

expr_test := \nabla_{\mu}{ A_{\alpha } } * \nabla^{\mu}{ A_{\beta} } + \nabla^{\mu}{ A_{\alpha } } * \nabla_{\mu}{ A_{\beta} };
sort_product(_);
sort_sum(_);
canonicalise(_);

At the end I still get:

\nabla_{\mu}{A_{\alpha}} \nabla^{\mu}{A_{\beta}}\discretionary{}{}{}
+
\nabla_{\mu}{A_{\beta}} \nabla^{\mu}{A_{\alpha}}

all indices are position=independent, but I tried switching to fixed and didn't help this particular issue.

Also, nabla is declared as:

\nabla{#}::Derivative.

But AFAIU the non-commutativity of Derivative should only affect order of indices, not the resulting object.

Rename specific dummy indices in expressions

Mon, 19 Jan 2026 19:20:52 +0000

In some cases, it would be useful to rename a specific subset of indices (dummy or not). In my particular case I am doing Hilbert metric variations of a matter Lagrangian and trying to reach a final expression that factors out a single metric variation coefficient.

What is happening is that I have many terms with metric variation factors, but each having different dummy indices. Even rename_dummies doesn't really help as it's a global action and I don't see a way to rename only the dummies of the \delta{g} terms.

What other strategy could be useful here? I'm thinking into a pseudo-code doing this:

for each \delta{g}^{?a ?b} term matching with a \delta{g} factor:
rename indices ?a to something unused or reserved in entire term
rename indices ?b to something unused or reserved in entire term

Issue with split_index and dummy index

Tue, 13 Jan 2026 11:06:33 +0000

I am new to Cadabra and have a couple of beginner questions that I haven’t been able to resolve on my own. I would greatly appreciate any help or clarification you can offer. I have the following Cadabra Code:

{M,N,P,Q,R#}::Indices(full, position=fixed).
{a,b,c,d,e#}::Indices(space1, position=fixed).
{a,b,c,d,e}::Integer(1..2).
{m,n,p,q,r#}::Indices(space2, position=fixed).

\partial{#}::PartialDerivative.
g_{M N}::Metric.
g^{M N}::InverseMetric.
g_{M?N?}::Symmetric.
g^{M?N?}::Symmetric.

Gtog:=\Gamma^{P?}_{M? N?}->(1/2)*g^{P? Q}(\partial_{N?}{g_{Q M?}}
+\partial_{M?}{g_{Q N?}}-\partial_{Q}{g_{M? N?}});

ex:=\Gamma^{2}_{1 2};
substitute(_, Gtog);
split_index(_, $M, a, m$);

todo:=\Gamma^{R}_{q m}\Gamma^{p}_{n R};
substitute(_, Gtog);

Question1: I tried using split index to split an index in my expression, but it doesn’t seem to do anything -- it simply returns the answer without performing the split. What might be the issue?

Question2: For the expression \Gamma^{R}{q m}\Gamma^{p}{n R} I am getting the same index Q repeated four times in the output. How can I prevent this and get the correct index contractions without the same dummy index appearing too many times?

absolute value in Cadabra expression

Fri, 19 Dec 2025 12:15:46 +0000

Hi, Sorry for the simplistic question, but what is the proper way to generate a Cadabra expression with absolute value?

I tried

ex := \sqrt{g} -> \abs(Z) \sqrt{h}

but I get an error, also using sympy.Abs doesn't seem to generate an output with |Z|

The mystery of meld

Wed, 19 Nov 2025 15:20:16 +0000

Hello,

Thanks a lot for Cadabra. This is very useful! But I found strange behavior of the algorithm meld.

First, a bug related to prefactor. Consider:

ex:= B^{i} B^{j} S^{m n} D_{i j} D_{m n} + B^{i} B^{j} S^{m n} D_{m n} D_{i j};
S^{i j}::Symmetric;
D_{i j}::Symmetric;

And then if I evaluate meld for the ex, I get prefactor 3/2 instead of 2.

Moreover, if I do not specify the symmetries of S^{i j} and D_{i j} separately, then the algorithm works correctly.

Second, an incorrect interpretation of an index convolution:

\partial{#}::PartialDerivative.
ex1:=  G^{m j} \partial_{i}{A} \partial_{m}{F_{j}} +  G^{m j} \partial_{m}{A} \partial_{i}{F_{j}};
ex2:= D^{m n} \partial_{i}{C} \partial_{m}{Q_{n j}} + D^{m n} \partial_{m}{C} \partial_{i}{Q_{n j}};
meld(ex1);
meld(ex2);

Meld interprets expressions as two identical terms in both cases, although I did not specify any symmetries of objects.

Thanks, Maksim

Strange behavior young_project_product with zoom

Wed, 22 Oct 2025 16:56:49 +0000

Hello, I found a peculiar effect of Young's projection when applied to part of an expression after using zoom. For example:

{a,b,c,d,f,e, m,n,k,l,q,r,s,t,u,v,w,x,y,z,m#, n#}::Indices(values={0..3}, position=independent);

\nabla{#}::Derivative.
h_{m? n?}::Metric.
h^{m? n?}::InverseMetric.
h_{m? n?}::Symmetric.
h^{m? n?}::Symmetric.
F_{a? b?}::AntiSymmetric.
F^{a? b?}::AntiSymmetric.
\nabla_{a}{F_{b c}}::TableauSymmetry(shape={2,1}, indices={0,1,2});
\nabla_{a}{\nabla_{b}{p}}::Symmetric;

and I have some long expression:

expr := 1/2 F_{a b} F_{c d} R \nabla_{e}(p) \nabla_{f}(p) h^{a c} h^{b e} h^{d f}
-F_{a b} F_{c d} \nabla_{e}(\nabla_{f}(p)) \nabla_{m}(\nabla_{n}(p)) h^{a c} h^{b e} h^{d f} h^{m n} 
+ F_{a b} F_{c d} \nabla_{e}(\nabla_{f}(p)) \nabla_{m}(\nabla_{n}(p)) h^{a c} h^{b e} h^{d m} h^{f n}
-F_{a b} F_{c d} R_{e f} \nabla_{m}(p) \nabla_{n}(p) h^{a e} h^{b m} h^{c f} h^{d n}
-2F_{a b} \nabla_{c}(F_{d e}) \nabla_{f}(\nabla_{m}(p)) \nabla_{n}(p) h^{a f} h^{b n} h^{c d} h^{e m} 
+ F_{a b} F_{c d} \nabla_{e}(\nabla_{f}(p)) \nabla_{m}(\nabla_{n}(p)) h^{a e} h^{b m} h^{c f} h^{d n} 
+ F_{a b} \nabla_{c}(F_{d e}) \nabla_{f}(\nabla_{m}(p)) \nabla_{n}(p) h^{a c} h^{b d} h^{e f} h^{m n};

And I want to simplify terms with \nabla{a}{F{b c}}:

zoom(expr, $Q?? \nabla_{a}{F_{b c}}$);
expr2 = take_match($@[expr2]$, $Q?? \nabla_{a}{F_{b c}}$);
canonicalise(young_project_product(expr2));

For expr I get - 4/3 F_{a b} \nabla_{c}(F_{d e}) \nabla_{f}(\nabla_{m}(p)) \nabla_{n}(p) h^{a f} h^{b n} h^{c d} h^{e m} but for expr2 I get

- 8/9 F_{a b} \nabla_{c}(F_{d e}) \nabla_{f}(\nabla_{m}(p)) \nabla_{n}(p) h^{a f} h^{b n} h^{c d} h^{e m}

vary behaves in an odd way

Mon, 06 Oct 2025 10:29:17 +0000

Here is a short example of what I found weird:

ex:= 2(r)**(-2) - B(r); vary(_, $B(r) -> \delta{B}$);

from which I get

2 {r}^{-2} - \delta\left(B\left(r\right)\right)

I tought that it was due to my older version of cadabra, but I just installed the latest version and the answer is the same. If I had in the first term just r, without an exponent, the answer woud be correct.

Thanks, Andrei

plot example on p. 51 of The Cadabra Book require range

Mon, 21 Jul 2025 15:05:29 +0000

A note on p. 52 tell the range is optional. But the example don't work without it.

Also, increasing the number of samples have no effect.

plot(sol, range_=($tau$, 0, 10),samples=5000, title="Damped coupled oscillators");

\dot command not displayed correctly

Mon, 14 Jul 2025 23:37:35 +0000

I'm trying to reproduce the first example on page 37 of The Cadabra Book.

I tried in cadabra2-gtk and jupyter kernel but I got a strange result. Instead of a dot in between I got a comma and a dot over.

In cadabra2-cli this is the output : \dot(A , D) \dot(B , C) which when entered in notebook give me the same wrong result.

Here is the code I'm running :


> {\mu, \nu, \rho, \sigma}::Indices(vector);

> ex:= A\\_\\{\mu\\} B\\_\\{\nu\\} C\\_\\{\nu\\} D\\_\\{\mu\\};

> substitute\(\_\, $A?\_\{\rho\} B?\_\{\rho\} \-\> \dot\{A?\}\{B?\}$\, repeat=True\);

Regex not working

Sat, 12 Jul 2025 21:06:37 +0000

I tried the example on page 37 of the Cadabra Book

ex:= A + B3 + C7;
substitute(_, $A + M? + N? | \regex{M?}{"[A-Z]7"} -> \sin(M? N?\)/N?$);

In jupyter I got this error :

Traceback (most recent call last):
  File "/usr/local/lib/python3.12/dist-packages/cadabra2_jupyter/kernel.py", line 57, in do_execute
    self._execute_python(pycode)
  File "/usr/local/lib/python3.12/dist-packages/cadabra2_jupyter/kernel.py", line 111, in _execute_python
    self._sandbox_context(pycode)
  File "/usr/local/lib/python3.12/dist-packages/cadabra2_jupyter/context.py", line 51, in __call__
    exec(code, self._sandbox)
  File "<string>", line 2
    substitute(_, ""Ex(r'''A + M? + N? | \regex{M?}{"[A-Z]7"} -> \sin(M? N?)/N?''', False)); display(_)
                  ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
SyntaxError: invalid syntax. Perhaps you forgot a comma?

The command line simply crash :

> ex:= A + B3 + C7; A + B3 + C7 > substitute(_\, $A + M? + N? | \regex{M?}{"[A-Z]7"} -> \sin(M? N?)/N?$);

Terminated by user interrupt.

I'm using Cadabra 2.5.12

How to remove/unset properties

Sat, 12 Jul 2025 17:26:32 +0000

I'm reading The Cadabra Book and try every example in the same cadabra session. I tried command line and Jupyter kernel and same result.

I got the following error because in the same session the first two lines interfere with the others :

> {m#, n#}::Indices(vector);
Property Indices(position=free) attached to {m#, n#}.
> {n, m, p, q}::Integer(1..d);
Property Integer attached to {n, m, p, q}.
> A_{m n p}::TableauSymmetry(shape={1,1}, indices={1,2});
Property TableauSymmetry attached to A_{m n p}.
> p = TableauSymmetry.get($A_{m n p}$)
> p.attach($B_{m n p}$)
> ex:= B_{m n p} - B_{m p n};
B_{m n p}-B_{m p n}
> canonicalise(_);
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
RuntimeError: Indicex has integer property, but explicit range needed.

Is there a way to remove properties ?

Thanks

execution problems

Tue, 08 Apr 2025 09:22:21 +0000

Hi Kasper, Thanks for the quick update. I've tried to run the latest version of cadabra2 but I get this run time error

Cadabra 2.5.11 (build 3796.22211b1d23 dated 2025-04-08)
Copyright (C) 2001-2025  Kasper Peeters <info@cadabra.science>
Traceback (most recent call last):
  File "<input>", line 1, in <module>
FileNotFoundError: [Errno 2] No such file or directory: '/Users/leo/tmp2//opt/homebrew/opt/python@3.13/Frameworks/Python.framework/Versions/3.13/lib/python3.13/site-packages/cadabra2_defaults.py'

I had no problems compiling the code. Here is what I did

git clone https://github.com/kpeeters/cadabra2
cd cadabra2/
git checkout devel
cmake -DENABLE_MATHEMATICA=OFF -DCMAKE_INSTALL_PREFIX:PATH=/Users/leo/tmp2/ ..
mkdir build
make install

export PATH=/Users/leo/tmp2/bin/:$PATH

I chose to install to ~/tmp2 just for testing. I've done this mant times before and b=nvere had this problem. Any thoughts? Cheers, Leo ps. Santa was very busy over Xmas so no surprise that a few gremlins got into the system :)

cadabra2python seg fault

Mon, 07 Apr 2025 04:21:01 +0000

Hi Kasper,

Just a short note. Running cadabra2python on any non-empty file produces a seg fault.

Cheers, Leo

simple substitute hits wrong target

Mon, 07 Apr 2025 04:09:04 +0000

Greetings,

Here is an interesting case.

This short code

tmp := 2 A B + 3 B C + 4 C D.
substitute (tmp, $A? -> \theta A?$)
collect_factors (tmp)

returns

tmp := 2 \theta^2 A B + 3 \theta^2 B C + 4 \theta^2 C D

while

tmp := 2 A B C + 3 B C D + 4 C D E.
substitute (tmp, $A? -> \theta A?$)
collect_factors (tmp)

returns

tmp := 2 \theta^3 A B C + 3 \theta^3 B C D + 4 \theta^3 C D E

Okay, so far , so good.

However, this code

tmp := 2 A + 3 B + 4 C.
substitute (tmp, $A? -> \theta A?$)

returns

tmp := 4 \theta A + 9 \theta B + 16 \theta C

Notice that the integer coefficients have been squared! That's not correct.

Quite interesting, n'est pas?

Cheers, Leo

curly-braces and lost indices

Mon, 07 Apr 2025 04:06:29 +0000

Greetings,

This short example

{a,b,c,d,e,f#}::Indices.

foo := A^{a} A^{b} g_{a b}.
substitute (foo,$A^{a}->B^{a}$)

does what you'd expect. It returns

foo := B^{a} B^{b} g_{a b}

But the following code

foo := A^{a} A^{b} g_{a b}.
substitute (foo,$A^{a}->{B}^{a}$)

returns

foo := B B g_{a b}

Now that seems wrong to me.

Any thoughts?

My guess is that the curly-braces around B in the second assignment disconnects B from the following indices. Those indices are orphaned and are thus lost in the substitution.

The simple answer is -- Don't use unnecessary curly-braces.

Cheers, Leo

Evaluate does not respect NonCommuting

Fri, 14 Mar 2025 07:08:52 +0000

Hi!

Thanks for Cadabra! I just started playing around. I'm using the Fedora 40 package of Version 2.5.10 (build private dated 2024-12-24).

I'm trying the following:

\partial{#}::PartialDerivative;
t::Coordinate;
O::Depends(t);
{H,O_0}::NonCommuting;

ex := O = \exp(i*H*t) O_0 \exp(-i*H*t);
ex2 := \partial_t{O};

substitute(ex2, ex);

So far so good. However

evaluate(_);

yiels 0 which should not be because H and O_0 are not commuting.

I've read that simplify should not be used (https://cadabra.science/qa/798/simple-commutator-in-quantum-mechanics?show=798#q798) but is this true for evaluate as well?

Thanks in advance,

Sebastian

How to manage two projects in their own folders which import .cnb's with the same names

Mon, 17 Feb 2025 07:24:51 +0000

I have two separate folders for different projects with some notebook (.cnb) files. In both folders I use the names methods.cnb, latex_macros.cnb, properties.cnb to split up the code, then import them into a main file to do work.

I noticed that, when importing from a .cnb, cadabra will convert it to a .py file and store it in the /home/user/cadabra_packages folder (on Ubuntu).

At the moment, I need to manually delete these cached files in cadabra_packages before I can swap between my projects.

I was wondering if there are any suggestions on how to make this better? I am very amateurish, but would appreciate a pointer in the right direction so that I can manage my files better. The only thing I would like is for it to be operating system agnostic, so I can also do it on mac.

Some ideas I thought of:

(1) Directly use .py files in the project folders so no .cnbs need to be cached.

(2) Rename the files e.g. methods1.cnb in the first folder, methods2.cnb in the second.

I am not keen on (2) because it would be nice to have the same layout in folders for different projects. I could compromise with (1), but I do enjoy editing the files using the notebook interface.

Thanks!

No GUI after compiling from the source (manjaro)

Thu, 02 Jan 2025 14:59:46 +0000

Happy New Year community!

I've been using manjaro linux as my main OS for four year now (perhaps even more than that), and I compile cadabra2 every few months. The standard process is:

cd ~/Software/git.src/cadabra2
git fetch
git pull
rm -Rf build/ && mkdir build && cd build && cmake ../ 
make -j6
sudo make install

After that, I can execute cadabra2-gtk without problems (most of the times, hehehe).

Lately, I haven't been able of using the cadabra notebook... The source is compiled, but when I call cadabra2-gtk I get the following:

(cadabra2-gtk:984345): Gtk-WARNING **: 11:47:21.946: Theme parsing error: gtk.css:68:35: The style property GtkButton:child-displacement-x is deprecated and shouldn't be used anymore. It will be removed in a future version

(cadabra2-gtk:984345): Gtk-WARNING **: 11:47:21.946: Theme parsing error: gtk.css:69:35: The style property GtkButton:child-displacement-y is deprecated and shouldn't be used anymore. It will be removed in a future version

(cadabra2-gtk:984345): Gtk-WARNING **: 11:47:21.946: Theme parsing error: gtk.css:70:34: The style property GtkCheckButton:indicator-size is deprecated and shouldn't be used anymore. It will be removed in a future version

(cadabra2-gtk:984345): Gtk-WARNING **: 11:47:21.946: Theme parsing error: gtk.css:71:36: The style property GtkCheckMenuItem:indicator-size is deprecated and shouldn't be used anymore. It will be removed in a future version

(cadabra2-gtk:984345): Gtk-WARNING **: 11:47:21.946: Theme parsing error: gtk.css:73:46: The style property GtkScrolledWindow:scrollbars-within-bevel is deprecated and shouldn't be used anymore. It will be removed in a future version

(cadabra2-gtk:984345): Gtk-WARNING **: 11:47:21.946: Theme parsing error: gtk.css:76:30: The style property GtkExpander:expander-size is deprecated and shouldn't be used anymore. It will be removed in a future version

(cadabra2-gtk:984345): Gtk-WARNING **: 11:47:21.949: Theme parsing error: gtk.css:83:29: The style property GtkStatusbar:shadow-type is deprecated and shouldn't be used anymore. It will be removed in a future version

which seems harmless... but also this:

cadabra-server: Symbol `PyInstanceMethod_Type' has different size in shared object, consider re-linking
cadabra-server: Symbol `PyCapsule_Type' has different size in shared object, consider re-linking
cadabra-server: Symbol `PyModule_Type' has different size in shared object, consider re-linking
cadabra-server: Symbol `PyType_Type' has different size in shared object, consider re-linking
cadabra-server: Symbol `PyByteArray_Type' has different size in shared object, consider re-linking
cadabra-server: Symbol `PyCFunction_Type' has different size in shared object, consider re-linking
cadabra-server: Symbol `PyProperty_Type' has different size in shared object, consider re-linking
cadabra-server: Symbol `PyBaseObject_Type' has different size in shared object, consider re-linking
cadabra-server: Symbol `PyFloat_Type' has different size in shared object, consider re-linking
*** stack smashing detected ***: terminated
terminate called after throwing an instance of 'std::logic_error'
  what():  Failed to read port from server.

At first the notebook opens, but after a couple of seconds it closes.

Question

What is happening to my cadabra installation?

derivatives are not evaluated in evaluation of components

Fri, 20 Dec 2024 13:12:42 +0000

Hi, I wanted to compute the metric of AdS{d+1} through its embedding in \mathbb{R}^{d+2}. When evaluating the components of the induced metric as below, it ends up with components that still contain derivatives, i.e. things like \partial{\phi}{\cos{\phi}}, whereas if I compute the derivative directly, e.g.

ll :=\partial_{\phi}{\cos{\phi}}
evaluate(ll);

it evaluates it correctly. So how do I evaluate these derivative if I have all the component together in an expression?

{t,\rho,\phi}::Coordinate;
{A,B}::Indices(higher,values={0,1,2,3}, position=fixed);
{\mu, \nu, \sigma, \kappa,\alpha,\beta,\gamma}::Indices(lower,values={t,\rho,\phi}, position=fixed);
X^{A}::Depends(t,\rho,\phi);

\partial{#}::PartialDerivative;

X^{A}::Depends(t,\rho,\phi,\partial_{\nu}{#});
g_{A B}::Metric(higher).
g^{A B}::InverseMetric(higher).

G_{\mu \nu}::Metric(lower).
G^{\mu \nu}::InverseMetric(lower).


metricRdp2 := [
g_{0 2} = 0, g_{0 3} = 0, g_{0 4} = 0,
g_{0 1} = 0, g_{1 2} = 0, g_{1 3} = 0,g_{1 4} = 0,
g_{2 3} = 0,g_{2 4} = 0,g_{3 4}= 0,  g_{4 4} = -1, g_{0 0} = -1, 
g_{1 1} = 1, g_{2 2} = 1, g_{3 3} = -1];

global2 := [ X^{0} = R \cosh{\rho} \cos{t}, 
X^{3} = R \cosh{\rho} \sin{t}, 
X^{1} = R \sinh{\rho} \sin{\phi},
X^{2} = R \sinh{\rho} \cos{\phi}];

cc := G_{\mu \nu} = \partial_{\mu}{X^{A}}\partial_{\nu}{X^{B}}g_{A B};

ex2 := G_{\mu \nu}:
substitute(ex2,cc);

evaluate(ex2)
canonicalise(ex2)
substitute(ex2,metricRdp2)
substitute(ex2,global2)
unwrap(ex2)
collect_factors(ex2);

compile failed using boost 1.87

Thu, 19 Dec 2024 23:01:07 +0000

Greetings good folk,

I just updated/upgraded HomeBrew on my Mac Silicon and now Cadabra fails to compile. I did a bit of digging and it seems that in moving from version 1.85 to 1.87 Boost has removed the asio::io_service stuff. It seems that this was marked as deprcated for some time but now they've pulled the pin.

I know next to nothing about C++ so I'm afraid I can't offer any fix. I did find a short discussion on stackoverflow here

https://stackoverflow.com/questions/59753391/boost-asio-io-service-vs-io-context

The errors pop up mostly in this file

cadabra2/libs/websocketpp/websocketpp/transport/asio/connection.hpp

Sorry to be the bearer of bad news in this festive season (:

Cheers, Leo

c++lib fails to compile

Tue, 10 Dec 2024 16:24:56 +0000

On my system, c++lib fails to compile properly. I needed to add the line ${CMAKE_CURRENT_SOURCE_DIR}/../core/properties/DerivativeOp.cc and also explicitly add the python include path, i.e. include_directories(/usr/include/python3.10)

I'm not sending a fix because my Python include is obviously system-dependent; probably there is a correct way to do this.

Kronecker delta wrong output in 2D

Tue, 10 Dec 2024 15:45:07 +0000

I'm basically following the eliminate_metric example from the website:

{m, n, p, q, r}::Indices(vector, position=fixed).
{m, n, p, q, r}::Integer(0..9).
g_{m n}::Metric.
g^{m n}::InverseMetric.
g_{m}^{n}::KroneckerDelta.
g^{m}_{n}::KroneckerDelta.
ex:=g_{m p} g^{p m};
eliminate_metric(_);
eliminate_kronecker(_);

but I'm working with doubled coordinates, meaning that my coordinates go from 0 to 2D-1. Therefore, if I change the corresponding part to

{m, n, p, q, r}::Integer(0..2*D-1).

the output is, against all odds and common sense, D. You may be thinking that it won't work with such terms, but I've stumbled upon a bizzare bug; here are some examples of different inputs and outputs:

{m, n, p, q, r}::Integer(0..D).

D+1

{m, n, p, q, r}::Integer(0..2 D + 1).

2D+2

{m, n, p, q, r}::Integer(0..3D-1).

D

{m, n, p, q, r}::Integer(0..2*(D-1)).

2D-1

It seems that when the dimensionality of Kronecker is a multiple of D, it stops working properly. Any ideas on why this is happening? What can be done to fix it?

How to handle metric on submanifold?

Sat, 07 Dec 2024 10:52:01 +0000

Hi,

I was trying to play with cadabra to study Einsteins equations at future null infinity. I have the coordinates $u,r,\phi,\theta$, for which I use lowercase greek indices ($\mu,\nu,...$) and I want to use the coordinate $\phi, \theta$ just to label points on the sphere, which I would like to label with uppercase latin indices (A,B,C,..)

Now I would like to define a metric just on the sphere, so I used in its definition only the indices A,B, hoping that it makes it an object only of those indices. Apparently this is not the case, since if I just evaluate the metric, I still get components written out for the u and r directions and if I try to complete the rule ssid, I get the error that the metric has zero determinant. Is there a way to make \gamma_{A B} an object only of the intrinsic coordinates \phi \theta such that e.g. complete works?

Here my minimal example:

{u,r,\phi,\theta}::Coordinate;

{\mu,\nu,\rho,\sigma}::Indices(higher,values={u,r,\phi,\theta}, position=fixed);
{A,B}::Indices(lower,values={\phi,\theta}, position=fixed,parent=higher);

\gamma_{A B}::Metric(lower, signature = 2);
\gamma^{A B}::InverseMetric(lower, signature = 2);
#\gamma_{A B}::Depends(r,\phi,\theta);

ssid:= {\gamma_{\phi \phi} = r**2*\sin(\theta)**2,   
       \gamma_{\theta \theta} = r**2
       };

ex1 := \gamma_{B A};
evaluate(ex1);
complete(ssid, $\gamma^{A B}$);

Canonicalise Kernel crash

Mon, 02 Dec 2024 19:13:19 +0000

Hello everyone.

I am currently calculating a large expression in coordinates and using the canonicalise function to order partial derivatives without using sympy bridge. And I found that an expression of this kind causes the kernel to crash when executing the canonicalise function.

{t, x, y, z, u}::Coordinate.
{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,v,w#}::Indices(full, values={t,x,y,z,u}, position=independent).
\partial{#}::PartialDerivative.

p::Depends(t).
W_{m?}::Depends(t,x,y,z).

test := Dh \partial_{x}(W_{x}) \partial_{y}(W_{y}) (\partial_{t}(p))**2 (\partial_{t t}(p))**2 (a)**(-4) a2X;

canonicalise(test);

I'm using Cadabra 2.5.8 (build 3421.a815a66da2 dated 2024-10-25) on Ubuntu 22.04.5

update:

I also noticed that if I write

canonicalise($\partial_{x}{W_{x}} \partial_{y}{W_{y} (\partial_{t}{p})**2 (\partial_{t t}{p})$);

I get the following error:

Traceback (most recent call last): File "/usr/local/lib/python3.10/dist-packages/cadabra2_jupyter/kernel.py", line 56, in do_execute self._execute_python(pycode) File "/usr/local/lib/python3.10/dist-packages/cadabra2_jupyter/kernel.py", line 110, in _execute_python self._sandbox_context(pycode) File "/usr/local/lib/python3.10/dist-packages/cadabra2_jupyter/context.py", line 51, in __call__ exec(code, self._sandbox) File "<string>", line 1, in <module> IndexError: vector::_M_range_check: __n (which is 3) >= this->size() (which is 3)

and If if I replace \partial_{t t}{p} with (\partial_{t t}{p})**2 , the kernel dies

canonicalise($\partial_{x}{W_{x}} \partial_{y}{W_{y} (\partial_{t}{p})**2 (\partial_{t t}{p})**2$);

collect_factors / simplify with AntiCommuting objects

Thu, 21 Nov 2024 20:35:21 +0000

This is a weird case and perhaps unlikely to be encountered in normal use, but I'm noting it here.

Consider:

{A,X}::AntiCommuting.
ex := X A X;
sort_product(ex)
collect_factors(ex);

The above produces a negative sign as one might suspect. However:

{A,X}::AntiCommuting.
ex := X A X;
collect_factors(ex);

The above does not. Nor does this:

{A,X}::AntiCommuting.
ex := X A X;
simplify(ex);

Cadabra in Jupyter notebook - how to store?

Sun, 20 Oct 2024 12:59:00 +0000

Hello, I'm working with Cadabra on Jupyter notebooks, specifically via JupyterHub running on a linux system. I save my work and shuts down the computer until my next session, but then when I open the notbook again I see that the results are fonw, only the code is saved. I think only once I managed to somehow save a notebook with all of the results still showing when I re-open it. I don't know how I did it.

So my questio nis how to save a Cadabra-notebook (on a jupyterhub) with all the data still in the cache, and saved even after the computer is shut down?

sympy(?) errors parsing nested derivatives?

Tue, 01 Oct 2024 06:47:52 +0000

As a warmup exercise before what I really want to do, I'm trying to compute the Laplacian operator in 2D plane polar coordinates. I do this by expressing \partial_{x} and \partial_{y} in terms of \partial_{r} and \partial_{\theta}, and then compute the sum of the squares of these operators.

Of course, operator expressions like \partial_x^2 are quickly regurgitated, so I'm trying the following alternative:

Px:= { px = \cos{\phi} \partial_{r}{f} 
     - \frac{\sin\phi}{r} \partial_{\phi}{f} } ;

Pxx:= { pxx = \cos{\phi} \partial_{r}{px} 
     - \frac{\sin\phi}{r} \partial_{\phi}{px}};

substitute(Pxx,Px);

evaluate(Pxx);

Cadabra gets the substitution correct, but evaluate() gives what looks like a sympy error:

{\color{red}{\begin{verbatim}TypeError: bad operand type for unary -: 'FunctionClass'

{\color{red}{\begin{verbatim}TypeError: bad operand type for unary -: 'FunctionClass'

TypeError: bad operand type for unary -: 'FunctionClass'
At:
  Notebook Cell (Line 0): 
  /tmp/.mount_CadabrIqWtS3/usr/lib/python3.8/dist-packages/sympy/parsing/sympy_parser.py(909): eval_expr
  /tmp/.mount_CadabrIqWtS3/usr/lib/python3.8/dist-packages/sympy/parsing/sympy_parser.py(1090): parse_expr
  Notebook Cell (Line 0):

Btw, I was unable to copy and paste these red error messages out of cadabra easily by the usual highlight-and-click methods. I had to click Edit->Copy and then paste it over here, but it still omitted the first line above (which I ended up typing in by hand).

\partial_\theta causes infinite loop(?) in server

Mon, 30 Sep 2024 08:53:35 +0000

As in my previous questions, I have

f::Depends(r,\theta,\phi, vr, vtheta, vphi);

The following works ok:

exfr:= { fr = \partial_r{f} };

but

exfth:= { fth = \partial_\theta{f} };

makes the server go into some kind of infinite loop (i.e., consumes ~ 99% CPU). So I have to restart the kernel.

Similarly,

exfvth:= { fvth = \partial_vtheta{f} };

causes infinite looping in server.

Enclosing the \theta or vtheta in braces works ok, i.e.,

exfth:= { fth = \partial_{\theta}{f} };

Gtk error messages on startup.

Mon, 30 Sep 2024 03:02:33 +0000

When I run Cadabra_2.5.6_x86_64.AppImage from a terminal I get the following error messages:

Gtk-Message: 12:54:41.239: Failed to load module "canberra-gtk-module" Gtk-Message: 12:54:41.264: Failed to load module "window-decorations-gtk-module" Gtk-Message: 12:54:41.264: Failed to load module "colorreload-gtk-module"

but it seems to keep running, although with fonts that are not my usual choice.

Also, (not necessarily at startup) I get messages on the terminal saying

cadabra-client: connection to server on port 33555 failed

Custom display symbols for variables?

Mon, 30 Sep 2024 02:37:00 +0000

As well as the usual spherical polar coordinates r,\theta,\phi, I also want to use their velocites as coordinates, which I'm currently representing as vr, vtheta, vphi. I'd prefer to display the latter as (latex) \dot r, \dot\theta, \dot\phi.

I vaguely recall cadabra1 had a way of doing this. Can I do it somehow in cadabra2?

Simple derivative evaluation not working for me.

Mon, 30 Sep 2024 02:27:19 +0000

So I copied and modified that start of schwarzschild.cnb to my own notebook which I'll call Laplacian.cnb

I modified the first code block to add some more coordinates vr, vtheta, vphi (intended to represent velocities -- I'm working in velocity phase space).

{t,r,\theta,\phi, vt, vr, vtheta, vphi}::Coordinate;
{\mu,\nu,\rho,\sigma,\lambda,\kappa,\chi,\gamma}::Indices(values={t,r,\phi,\theta}, position=fixed);
\partial{#}::PartialDerivative;
g_{\mu\nu}::Metric.
g^{\mu\nu}::InverseMetric.

Now I try to define a simple (scalar) function f, which is just f() = r + vtheta**2 and try to evaluate \partial_r f. My code block is

f::Depends(r,\theta,\phi, vr, vtheta, vphi);
exf:= { f = r + vtheta**2 } ;
exfr:= { fr = \partial_r{f} };
evaluate(exfr,exf);

The final line of output is just \partial_r f. I.e., it didn't perform the partial derivative. (The answer should have been 1.)

What am I doing wrong?

Another "No module named 'sympy' " question.

Sun, 29 Sep 2024 08:23:23 +0000

After a long absence I've decided to try and restart using Cadabra again.

I'm running OpenSUSE (Leap) for which there doesn't seem to be a recent build on github, so I downloaded Cadabra_2.5.4_x86_64.AppImage instead.

Then I tried to run the schwarzschild.cnb tutorial. At the 2nd code block it complains that it cannot find the sympy module.

I reinstalled Anaconda, then used conda to install and update python3 and sympy. This didn't help. I also tried "pip install sympy", but this didn't help either.

What am I doing wrong?

[BTW: if I run python3 standalone, it imports sympy without complaint.]

Cycle freeze

Thu, 26 Sep 2024 15:31:48 +0000

Hi!

I often use the following function with a converge loop to simplify expressions:

def tidy (expr):
   converge(expr):
       distribute(expr)
       product_rule(expr)
       sort_product(expr)
       collect_factors(expr)
   return expr

But after today's update of cadabra to the latest version, this loop became infinite. Not for all expressions, but, for example, for this one

{a,b,c,d,f,g,m,n,k,q,r,s,t,u,v,w,x,y,z,m#}::Indices(subspace, position=independent, parent=full);
\partial{#}::PartialDerivative.


expr := ((\phi)**(-2) + A_{a} h^{a b} A_{b}) (\partial_{n}((\phi)**2 A_{m}) + \partial_{m}((\phi)**2 A_{n}));

Also, the problem only occurs if I use this function from a external file

import functions as fun

so, if I use

fun.tidy(expr)

Loop is infinite.

Expanding sum into time and purely space components.

Sat, 10 Aug 2024 12:23:25 +0000

Greetings.

Ricci tensor perturbations involve sum over spacetime components (in FLRW ->{t, r, \theta \varphi} ). How to expand spacetime sum (greek indices) into time and purely space components (latin indices)?
For each Gamma and \delta{\Gamma} there are subrules (see related image): Gamma^{t}_{t t}= ... Gamma^{t}_ {t i}=... How to setup and substitute this relations in master equations?

The core of programm is

import cdb.core.manip as mp
import cdb.core.component as cmp
{t,r, \theta, \varphi}::Coordinate.
a::Depends(t).
{\mu, \nu, \rho, \sigma, \alpha, \beta, \gamma, \lambda, \tau, \chi, \iota, \upsilon, \omega}::Indices(curved,position=fixed,values={t,r, \theta, \varphi}).
{a,b,c,d,e,f,g,i,j,k,l,m,n,p,s}::Indices(position=fixed,values={r, \theta, \varphi})
gb{#}::LaTeXForm("\bar{g}").
gb_{\mu \nu}::Symmetric.
gb::Diagonal;
{gb^{\mu}_{\nu}, gb_{\mu}^{\nu}}::KroneckerDelta.
back := gb\_\{t t\}=\-1\, gb\_\{r r\}=a**2/(1-k*r**2), gb_{\theta \theta}=a**2*r**2, gb_{\varphi \varphi}=a**2*r**2*\sin{\theta}**2, gb^{t t}=-1, gb^{r r}=(1-k*r**2)/(a**2), gb^{\theta \theta}=1/(r**2_a**2), gb^{\varphi \varphi}=1/(a**2_r\*\*2\*\sin\{\theta\}\*\*2\)\, gb\_\{t r\}=0\, gb\_\{t \theta\}=0\, gb\_\{t \varphi\}=0\, gb\_\{r \theta\}=0\, gb\_\{r \varphi\}=0\, gb\_\{\theta \varphi\}=0\, gb\_\{r t\}=0\, gb\_\{\theta t\}=0\, gb\_\{\varphi t\}=0\, gb\_\{\theta r\}=0\, gb\_\{\varphi r\}=0\, gb\_\{ \varphi \theta\}=0;

In the part we define components of ricci perturbations:

gs{#}::LaTeXForm("\overline{\gamma}")
dGamma{#}::LaTeXForm("\delta\Gamma");

dRic{#}::LaTeXForm("\delta R").
perric := \delta{R_{\mu \nu}} = \partial_{\sigma}{dGamma^{\sigma}_{\mu \nu}} - \partial_{\nu}{dGamma^{\sigma}_{\mu \sigma}} + dGamma^{\rho}_{\mu \nu} \Gamma^{\sigma}_{\rho \sigma} + \Gamma^{\rho}_{\mu \nu} dGamma^{\sigma}_{\rho \sigma} - dGamma^{\rho}_{\mu \sigma} \Gamma^{\sigma}_{\nu \rho} - \Gamma^{\rho}_{\mu \sigma} dGamma^{\sigma}_{\nu \rho};

dr1 = cmp.get_component(perric, $t,t$);
dr2 = cmp.get_component(perric, $t,i$);
dr3 = cmp.get_component(perric, $j,k$);

How to expand sum?

We try to define relation for non perturbed Gamma!:

eq1 := {\Gamma}^{t}_{i j}=(a \partial_{t}{a} {gs}_{i j});

But error appear: RuntimeError: Free indices on lhs and rhs do not match.

Thank you in advance for your help.

question about eliminate_metric

Fri, 02 Aug 2024 14:27:21 +0000

For example,

{\mu,\nu,\rho}::Indices(vector).
g_{\mu\nu}::Metric.
g^{\mu\nu}::InverseMetric.
g_\mu^\nu::KroneckerDelta.
g^\mu_\nu::KroneckerDelta.
\delta{#}::Accent.

ex:=\delta{g_{\mu\nu}} g^{\nu\rho};
eliminate_metric(_);

I get an odd result.

A bug of Accent

Thu, 01 Aug 2024 04:36:53 +0000

For example,

{a,b}::Indices(vector).
\delta{#}::Accent.
ex:=\delta{A^{a b}+B^{a b}}

Result is not right.

I'm trying to reproduce equation 4.6 from 4.5 in Carrolls Spacettime ;and Geometry

Thu, 27 Jun 2024 22:01:26 +0000

The following is my code. The last line has a partial with respect to nu and I want to use the chain rule to convert to d/dlamda.

I'm an newbie here so I may be doing something reallty dumb. I did convert the partial declaration to have the Derivative property so I don't end up with a double partial derivative of \nu and \lambda. Any suggestions would be appreciated.


{t,x,y,z,\lambda}::Coordinate.
{t,x,y,z,}::Depends(\lambda);
{\mu,\nu,\sigma,\rho,\alpha}::Indices(values={t,x,y,z},position=independent);
\nabla{#}::Derivative.
#\partial{#}::PartialDerivative;
\partial{#}::Derivative;

#deriv := \nabla_{\mu}{v?^{\nu}} -> \partial_{\mu}{v?^{\nu}} + \Gamma^{\nu}_{\sigma \mu} v?^{\sigma};

deriv := \nabla_{\nu}{\partial_{\lambda}{x^{\mu}}} -> \partial_{\nu}{\partial_{\lambda}{x^{\mu}}} 
    + \Gamma^{\mu}_{\sigma \nu} \partial_{\lambda}{x^{\sigma}};

expr := \partial_{\lambda}{x^{\nu}} * \nabla_{\nu}{\partial_{\lambda}{x^{\mu}}};

substitute(expr,deriv);

distribute(expr);
sort_product(expr);

Factor out of numerical coefficient

Mon, 24 Jun 2024 13:41:15 +0000

Hi everyone! I'm using cadabra for a several years, but still can't understand how to factor out numerical coefficients, for example

factor_out($6 A B + 6 A C$, $A$);

factor_out($6 A B + 6 A C$, $A,6$);

It would be much prettier if you could factor out the numerical coefficient too. it doesn't affect the functionality, but when exporting formulas to latex, a lot of time is spent on manual editing.

how to deal with lovelock gravity?

Sun, 23 Jun 2024 03:15:45 +0000

For Lovelock gravity,

$$ L=R_{a_{2}b_{2}}^{c_{2}d_{2}}\cdots R_{a_{m}b_{m}}^{c_{m}d_{m}}\delta_{c_{2}d_{2}\cdots c_{m}d_{m}}^{a_{2}b_{2}\cdots a_{m}b_{m}} $$

where

$$ \delta_{{jc_{1}d_{1}...c_{m}d_{m}}}^{{ia_{1}b_{1}...a_{m}b_{m}}}=\det\left[\begin{array}{c|ccc}\delta_{j}^{i} & \delta_{{c_{1}}}^{i} & \cdots & \delta_{{d_{m}}}^{i}\\hline \\delta_{j}^{{a{1}}}\\vdots & & \delta_{{c_{1}d_{1}...c_{m}d_{m}}}^{{a_{1}b_{1}...a_{m}b_{m}}}\\delta_{j}^{{b\{m}}}\end{array}\right] $$

when $m=2$, we get Einstein gravity; when $m=3$, we get Gauss-Bonnet gravity. But how to deal with it more generally? More concretely, I don't know how to do it use for loop.

LaTeXForm fails in version 2.5.2

Mon, 17 Jun 2024 22:50:17 +0000

Greetings,

This simple statement

Dx{#}::LaTeXForm{"Dx"}.

raises a syntax error

SyntaxError: '{' was never closed

I'm running Cadabra version 2.5.2. build 3043.2a1d62739a.

Also, this version of Cadabra compiles fine under MacOS Sonoma 14.5.

Cheers, Leo

substitute gives incorrect value on scalar expression

Mon, 17 Jun 2024 22:25:46 +0000

Greetings, I'm trying to use the substitute algorithm to set the value of a scalar expression but I'm getting some very odd results. In the following examples I set a variable s to be 1 and the expected result should be zero. But in the first case I get 2. This problem only seems to arise for the single choice of s=1. Cheers, Leo

# ----------------------------------------------------
# fails, should return 0

foo := s - s**3;

substitute (foo, $s = 1$);

# ----------------------------------------------------
# success

foo := s - s*s*s;

substitute (foo, $s = 1$);

# ----------------------------------------------------
# success

foo := s**3 - s;

substitute (foo, $s = 1$);

# ----------------------------------------------------
# success

foo := s - s**3;

substitute (foo, $s = 2$);

# ----------------------------------------------------
# success

foo := s - s**3;

substitute (foo, $s = 1/2$);

Cadabra 2.5.x released

Sat, 15 Jun 2024 19:27:27 +0000

I have today taken Cadabra 2.5.x out of experimental mode and made it the default.

This version contains built-in code to handle LaTeX typesetting, making the notebook interface much faster and enabling things like dynamical cell updates (e.g. updating counters) as well as full UTF-8 support for non-English languages (and emojis...). In the process, I have also fixed a number of issues with the notebook, e.g. interrupting calculations.

As a bonus, it is now possible to distribute Cadabra as an AppImage on Linux, which installs on any reasonably modern distribution. Hopefully this will help to reach a larger audience. I have also updated the homebrew package for macOS. Standard .deb packages for Ubuntu 22.04 are available as well.

Any feedback is welcome; please try this one and let me know!

location of bracket blocks distribute

Tue, 11 Jun 2024 05:46:21 +0000

Hi Folks,

Here are two instances of the distribute algorithm. Only the first gives the expected result. The only difference in the code is where the opening bracket is placed.

# ----------------------------------------------
# this works as expected

foo := A (
       B + C);

distribute (foo);

# ----------------------------------------------
# this fails to apply the "distribute" algorithm

bah := A
       (B + C);

distribute (bah);

Cheers, Leo

leading scalar expression blocks canonicalise

Tue, 11 Jun 2024 05:35:33 +0000

Hi Troops,

Here is s short Cadabra code

{a,b,c,d,e,f#}::Indices(position=independent).

R_{a b c d}::RiemannTensor.

# this should give zero
foo := (1+s) R_{a b c d} p^{a} p^{b} p^{c} p^{d}.
canonicalise (foo);

# this gives the expected result
bah := s R_{a b c d} p^{a} p^{b} p^{c} p^{d}.
canonicalise (bah);

There are two ouputs foo and bah. Both should be zero (beacuse of the symmetries of the Riemann tensor). But only bah is correctly given as zero. This seems like a bug to me.

Cheers, Leo

How to copy output as tex format in jupyter notebook?

Mon, 27 May 2024 16:38:47 +0000

Today, I try cadabra2 in jupyter, and it's very nice, but I don't know how to do copy output as tex format.

Update: When I right-click the mouse, I can see mathjax copy latex, but this just works when I run cell, and it just can copy the final result.

imp package was removed from python 3.12

Sat, 18 May 2024 18:30:19 +0000

Hi! I've find out that imp package was removed from python 3.12, and I can't import cdb.core.manip any more because the following error appears: ModuleNotFoundError: No module named 'imp' imp module is deprecated in favour of importlib.

Why does evaluate of metric eating vector shows only non-zero components.

Tue, 14 May 2024 20:17:56 +0000

I do not understand why g(.,v) only shows non-zero components after evaluation.

{m,n,a,b,c,r,s}::Indices(values={0,1,2,3}).
#
g_{a b}::Metric.
g^{a b}::InverseMetric.
#
g:=[g_{0 0}=-1, g_{0 1}=0, g_{0 2}=1, g_{0 3}=0, g_{1 0}=0,  g_{1 1}=1, g_{1 2}=0, g_{1 3}=0, g_{2 0}=1, g_{2 1}=0, g_{2 2}=1, g_{2 3}=0, g_{3 0}=0, g_{3 1}=0, g_{3 2}=0, g_{3 3}=1].
v:=[v^{0}=1, v^{1}=1, v^{2}=1, v^{3}=1];
display(v)
ex:=g_{a b} v^{a};
evaluate(ex,join(g,v), rhsonly=True);

OUTPUT gives
□_{b}
    {1} = 1
    {2} = 2
    {3} = 1

Setting g00=1 (tather than -1 above) gives

□_{b}
    {0} = 2
    {1} = 1
    {2} = 2
    {3} = 1

How can I store the result of evaluating g(.,v) as a co-vector?

Tue, 14 May 2024 15:40:59 +0000

Mathematically I expect g(.,v) to be a co-vector. Now, given:

import cdb.core.component as comp
{m,n,a,b,c,d,q,r,s}::Indices(values={0,1}).
g_{a b}::Metric.
g^{a b}::InverseMetric.
g:=[g_{0 0}=28, g_{0 1}=2, g_{1 0}=3, g_{1 1}=-1];
v:=[v^{0}=12, v^{1}=-11];
ex:=g_{a b} v^{a};
gv:=evaluate(ex,join(g,v), rhsonly=True)
w:=w_{a};

I would like to loop through components of gv and assign component values produce the desired co-vector w:

w_{b}=comp.get_component(w, b);

I have tried many approaches which fail.

Christoffel symbols computation in $d$ dimensions with split indices

Tue, 14 May 2024 14:17:33 +0000

I have a metric of the form, g_{\mu \nu} =

0	0	1
0	b_{ij}	u
1	u^T	f

where $u$ and $b_{i j }$ do not depend on the first coordinate $s$. The inverse metric is g^{\mu \nu} =

b^{i j} u_i u_j - f	- u_i b^{i j}	1
-b^{i j}u_j	b^{i j}	0
1	0	0

I would like to compute the Christoffel symbols I first try to followed section 2.5 in the paper, https://arxiv.org/pdf/hep-th/0701238

After the comments I have updated my code, here is the edited code and an explanation.

I wanted to split the Greek indices (${ \mu, \nu ...}$) into three parts: first: a capital Latin index that runs from 2 to $d$, second: small latin letters that run from 2 to $d-1$ and the last index $s1, s2,s3$ to the last coordinate $d$. To simplify the problem I tried first $d=4$ so for example, $\mu=1,2,3,4$, $A=2,3,4$, $i,j = 2, 3$ and $s1=4$. To do this splitting I used,

{\mu,\nu,\rho,\sigma,\kappa,\lambda,\eta,\chi#}::Indices(full, position=fixed, values={1,2,3,4}).
{A,B,C,D}::Indices(subspace, position=fixed, parent=full, values={2,3,4}).
{i,j, m,n,p,q,r,v#}::Indices(subspace2, position=fixed, parent=subspace, values={2,3}).
{s1,s2,s3}::Indices(subspace1, position=fixed, parent=subspace, values={4}).

the updated code following the sugestions in the comments is,

\partial{#}::PartialDerivative.
g_{\mu\nu}::Metric.
g^{\mu\nu}::InverseMetric.
g_{\mu? \nu?}::Symmetric.
g^{\mu? \nu?}::Symmetric.

b_{i j}::Metric.
b^{i j}::InverseMetric.
\delta^{\mu?}_{\nu?}::KroneckerDelta.
\delta_{\mu?}^{\nu?}::KroneckerDelta.

Gtog:= \Gamma^{\lambda}_{\mu \nu} -> (1/2) * g^{\lambda \kappa} (\partial_{\nu}{ g_{\kappa\mu}} + \partial_{\mu}{g_{\kappa \nu}} + \partial_{\kappa}{g_{\mu\nu} } );

Gamma4ij := \Gamma^{4}_{i j};

substitute(Gamma4ij, Gtog);

split_index(Gamma4ij, $\mu, 1, A$, repeat=True);

split_index(Gamma4ij, $A, i, 4$, repeat=True);

substitute(_, $g_{1 m} -> 0$ )
substitute(_, $g_{1 4} -> 1$ )
substitute(_, $g_{m 1} -> 0$ )
substitute(_, $g_{m n} -> b_{m n}$ )
substitute(_, $g_{m 4} -> U_{m}$ )
substitute(_, $g^{4 1} -> 1$ )
substitute(_, $g_{4 m} -> U_{m}$ )
substitute(_, $g_{4 4} -> f$ )

substitute(_, $g^{1 1} -> U_{m} b^{m n} U_{n} - f$ )
substitute(_, $g^{1 m} -> - U_{n} b^{n m}$ )
substitute(_, $g^{1 4} -> 1$ )
substitute(_, $g^{m 1} -> - U_{n} b^{n m}$ )
substitute(_, $g^{m n} -> b^{n m}$ )
substitute(_, $g^{m 4} -> 0$ )
substitute(_, $g^{4 1} -> 1$ )
substitute(_, $g^{4 m} -> 0$ )
substitute(_, $g^{4 4} -> 0$ )

substitute(_, $\partial_{1}{U??} -> 0$, repeat=True)
substitute(_, $\partial_{1 \mu?}{U??} -> 0$, repeat=True)
substitute(_, $\partial_{\mu? 1}{U??} -> 0$, repeat=True)
substitute(_, $\partial_{\mu? 1}{b???} -> 0$, repeat=True)

canonicalise(_);

I obtained the output $\Gamma^{4}_{i j} $, while if I change the definition of the Christoffel symbols and include ?,

Gtog:= \Gamma^{\lambda?}_{\mu?\nu?} ->
 (1/2) * g^{\lambda?\kappa} (
 \partial_{\nu?}{ g_{\kappa\mu?} } + \partial_{\mu?}{g_{\kappa\nu?} }
- \partial_{\kappa}{ g_{\mu?\nu?} } );

I obtain 0, but I know $\Gamma^{4}$ is not 0 so I don't know where my code is wrong. I would like to ask, how can I improve it and how can I generalise it to $d$ dimensions.