Cadabra Q&A - Recent questions in General questions

Tutorial on Differential Forms: I would expect A1 ^ A2 + A2 ^ A1 = 0

Sat, 02 May 2026 18:01:24 +0000

Hi, I am just beginning. Looking at the tutorial I am confused by the answer cadabra gives: A1 ^ A2 + A2 ^ A1 = 2 A1 ^ A2. I would have expected 0 as result.

Problem with SympyBridge

Mon, 13 Apr 2026 14:22:18 +0000

I am attempting to use SympyBridge to simplify a somewhat involved algebraic expression and it appears to be throwing a python error. Here is the notebook

https://drive.google.com/file/d/16jcPyMk7OkSsiShWLiYIw_7ng7-MUdFG/view?usp=sharing

The error message after the call to SympyBridge is

\begin{verbatim}TokenError: ('unexpected character after line continuation character', (1, 82))

At: /usr/lib/python3.12/tokenize.py(582): _generate_tokens_from_c_tokenizer /usr/lib/python3/dist-packages/sympy/parsing/sympy_parser.py(891): stringify_expr /usr/lib/python3/dist-packages/sympy/parsing/sympy_parser.py(1072): parse_expr Notebook Cell (Line 1): ex_sympy = sbb.to_sympy() \end{verbatim}

I have used SympyBridge on simpler expressions like those in the Cadabra Book and it works fine....

Trouble running standard cnb file

Wed, 01 Apr 2026 16:10:51 +0000

I have installed cadabra2 on a Ubuntu 24.04 system using the deb file provided. I imported the sample notebook called einstein_equations.cnb which derives Einstein's equation using an action variation. The last cell to be executed using python gives the following error:

substitute: Index error in replacement rule. substitute: No numerical pre-factors allowed on lhs of replacement rule.

The offending cell is

t1 = action[1][0][0][2]
eom:= 2\kappa @(t1) = 0;
distribute(_)
collect_factors(_)
manip.to_rhs(_, $- \kappa T_{\mu\nu}$);
substitute(_, manip.swap_sides(einsteinTensor));

and the last substitute command appears to give the error. manip.swap_sides command is coming from the imported cdb.core.manip package.

Any ideas what is going wrong?

Also could anyone enlighten me on the syntactical meaning of the first two lines of this cell. I couldn't find any references to it in the manual...

Thanks

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|

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

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!

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);

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, $\gamma^{A B}$);

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?

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?

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.

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.

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.

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.

why can't cadabra find cdb.sympy.calculus?

Tue, 07 May 2024 20:19:25 +0000

Running only the below produces no result ex := x+y; diff( ex, $y$ );

try to import

from cdb.sympy.calculus import diff ex := x+y; diff( ex, $y$ );

produces...

from cdb.sympy.calculus import diff

ModuleNotFoundError: No module named 'cdb.sympy.calculus'

how to compute metric(vector,vector) or g(v,w) where g,v,w already defined without haqving to repeat the definitions.

Tue, 07 May 2024 14:31:01 +0000

I am able to compute g(v,v) but have to repeat the definition of the component definitions (values or functions). Below is the is the code that "works" but it is strange that I have to repeat the definition of g and v in the line

evaluate(ex, $g_{0 0}=28, g_{0 1}=2, g_{1 0}=3, g_{1 1}=-1, 
    v^{0}=12, v^{1}=-11 , v^{0}=12, v^{1}=-11$, rhsonly=True)

I want to write something like

evaluate(ex, [g, v])

or prefereably

evaluate(g(v,v))

 ############################ 
 #CODE

{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}.
display(g)
display("......................0")
v:={v^{0}=12, v^{1}=-11}.
display(v)
display("......................1")
ex:=g_{a b} v^{b}.
display(ex)
display("......................2")
evaluate(ex, $v^{0}=12, v^{1}=-11$, rhsonly=True)
display(ex)
display("......................3")
ex:=g_{a b} v^{b}.
evaluate(ex, $g_{0 0}=28, g_{0 1}=2, g_{1 0}=3, g_{1 1}=-1, v^{0}=12, v^{1}=-11$, rhsonly=True)
display(ex)
display("......................4")
ex:=g_{a b} v^{a} v^{b}.
display(ex)
display("......................5")
evaluate(ex, $g_{0 0}=28, g_{0 1}=2, g_{1 0}=3, g_{1 1}=-1, v^{0}=12, v^{1}=-11 , v^{0}=12, v^{1}=-11$, rhsonly=True)
display(ex)
display("......................6")

convert other format to cadabra2 format

Wed, 24 Apr 2024 13:31:05 +0000

From Using Cadabra and Sympy together, I learn how to convert cadabra2 format to sympy format, but how to do the reverse operation? I.e. How to convert other formals, e.g. sympy format, to cadabra2 format?

question about keep_weight

Fri, 19 Apr 2024 03:46:06 +0000

In the following code,

{u,v,x}::Coordinate.
f::Coordinate.
{a,b,c,d,e,g,h,p,r#}::Indices(full,values={u,v,x},position=independent).
\partial{#}::PartialDerivative.
\nabla{#}::Derivative.
g_{a b}::Metric.
g^{a b}::InverseMetric.
g_a^b::KroneckerDelta.
g^a_b::KroneckerDelta.
g^{a b}::Symmetric.
g_{a b}::Symmetric.
{n_u,n_v,\gamma,w,\alpha,\lambda,A,B,C}::Depends({u,v,f,x}).

{u,v}::Weight(label=field,value=1);
{f}::Weight(label=field,value=2);

metric:={g_{u v}=1,
         g_{v u}=1,g_{v v}=-u**2\alpha,g_{v x}=u w,
         g_{x v}=u w,g_{x x}=\gamma};   
complete(_,$g^{a b}$);

from cdb.sympy.calculus import *

lamb:=(g^{a b}\partial_a{u v}\partial_b{u v})**{-1/2} ;
evaluate(_,metric,rhsonly=True);
substitute(_,$v->f/u$)
simplify(_);
substitute(_,$\alpha->a,w**2\gamma**{-1}->b$);
series(_,$f$,0,5/2);
substitute(_,$\bigO(A??)->0$)
keep_weight(_,$field=1$);

The final result is not right. There seems to be something wrong here.

function expand_power not work for fractional power

Thu, 18 Apr 2024 13:20:11 +0000

For example,

ex:=(a b)**{1/2};
expand_power(_);

Maybe some function like Assumption in Mathematica can be added.

How to calculate Christoffel symbols with metric defined by abstract matrices?

Tue, 16 Apr 2024 13:33:31 +0000

For example, when looking at ADM formalism, I have $ds^2 = -N^2dt^2+g_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt)$ in which the specific function relation between $g_{\mu \nu}$ and coordinates is not that clear so is abstract. Then how to do calculation on this kind of metric to get, say, the Christoffel symbols or other quantities after that? Below is what I've tried, but it didn't work with a result that all the Christoffel symbols are 0.

By the way, as a very beginner in ADM formalism, is it the case that I have some misunderstanding in the calculation of it?

{t,x,y,z}::Coordinate;
{\mu,\nu,\rho,\sigma,\lambda,\kappa,\chi,\gamma,i,j,k}::Indices(values={t,x,y,z},position=fixed);
\partial{#}::PartialDerivative;
g_{\mu\nu}::Metric;
g^{\mu\nu}::InverseMetric;
h_{i j}::Metric;
h^{i j}::InverseMetric;
ss:={ g_{t t}=0,
    g_{0 i}=N_i,
    g_{i j}=h_{i j}
};
complete(ss, $g^{\mu\nu}$);
ch:= \Gamma^{\mu}_{\nu\rho} = 1/2 g^{\mu\sigma} ( 
                                   \partial_{\rho}{g_{\nu\sigma}} 
                                  +\partial_{\nu}{g_{\rho\sigma}}
                                  -\partial_{\sigma}{g_{\nu\rho}} );
                                  evaluate(ch, ss, rhsonly=True);

Error printing as mma format

Fri, 12 Apr 2024 23:19:59 +0000

When I print the following expression as mma format $$ v {\left(\sqrt{f \left(f \alpha \gamma_{x x} \gamma_{y y}-f \alpha {\gamma_{x y}}^{2}+f \gamma_{x x} {w_{y}}^{2}-2f \gamma_{x y} w_{x} w_{y}+f \gamma_{y y} {w_{x}}^{2}-2\gamma\right) {\gamma}^{-1}}\right)}^{-1} $$ using code:

print(_.mma_form())

I got the following error message:

LaTeX error, probably a missing style file. See the output below.

This is pdfTeX, Version 3.141592653-2.6-1.40.22 (TeX Live 2022/dev/Debian) (preloaded format=pdflatex)
 restricted \write18 enabled.
entering extended mode
(./fileKtiEnn.tex
LaTeX2e <2021-11-15> patch level 1
L3 programming layer <2022-01-21>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2021/10/04 v1.4n Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size11.clo))
(/usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty
(/usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty))

Package geometry Warning: `lmargin' and `rmargin' result in NEGATIVE (-266.7303
6pt).
    `width' should be shortened in length.


Package geometry Warning: `tmargin' and `bmargin' result in NEGATIVE (-7740.856
78pt).
    `height' should be shortened in length.

How can I solve this problem?

Question about Gaussian normal coordinate

Thu, 11 Apr 2024 07:20:30 +0000

When I do calculation under Gaussian normal coordinate:

$$ d s^2=-2 d u d v-u^2\alpha(x) d v^2-2u w_i(x) dv dx^i+\gamma_{ij}(x) d x^i d x^j\,\quad x=(uv\,,x^i) $$

How to get inverse metric and others directly? I don't know how to deal with $w_i$ and $\gamma_{ij}$.

Formula rendering in jupyter notebook 7 and jupyter lab

Thu, 04 Apr 2024 13:24:21 +0000

Hi! I recently updated jupyter notebook and jupyter lab, and found out that the latest versions use mathjax3 instead of mathjax2 to render formulas. Their developer promises 80% faster rendering and it's really cool, except that the formulas are now rendered as one line, one very long line. Maybe someone has already dealt with that and managed to fix it?

how to swap indices for partial derivatives

Tue, 19 Mar 2024 09:51:53 +0000

I need Cadabra to understand that the indices of derivatives can be swapped. For example, part of my expression looks like this:

2 \chi w**(3) M**(\xi+1) \delta{z} 
    \partial_{a b}{z} \partial_{a c d}{z} v_{b} v_{c} v_{d} - 
2 \chi w**(3) M**(\xi+1) \delta{z} 
    \partial_{a b}{z} \partial_{c d a}{z} v_{b} v_{c} v_{d}

In theory, these 2 terms should contract, but they do not.

The indices are given as:

{a,b,c,d,f,i,j,h,k,l,m,n,p,r,s,t,u,v,w,z}::Indices("flat",position = free).
{a,b,c,d,f,i,j,h,k,l,m,o,p,r,s,t,u,v,w,z}::Integer(1..N) .

Partial derivative as

\partial{#}::PartialDerivative.

Hello i am a beginner in using Cadabra 2

Sat, 16 Mar 2024 19:44:04 +0000

iam facing a repeated question , "index error in replacement rule", i want to know the reasons for this error?

How to create derivatives of coordinates with indices?

Thu, 15 Feb 2024 17:28:42 +0000

I have two sets of coordinates, and two partial derivatives which can act on these coordinates. These should work in a way such that:

$$ \partial_{x^{\mu}} x^{\nu} = \delta_{\mu}^{\nu} = \partial_{a{\nu}} a\{\mu}. $$

How would I go about implementing this in Cadabra? I've tried defining partial derivatives of corresponding coordinates directly, I've tried using notation like DX and DA to differentiate between the two and then using LaTeXform to make it more tidy but I inadvertently run into a stop of some sort.

Any and all help will be greatly appreciated!

derivatives with two indices

Thu, 18 Jan 2024 10:13:42 +0000

Hello,

I need to use derivatives with two indices. However, when I use product_rule or integrate_by_parts, Cadabra splits the derivatives into two derivatives with one index each. Is there any way to circumvent this ?

Thank you

How do I use object properties in a python executable?

Sat, 06 Jan 2024 11:28:02 +0000

Hello, I am trying to declare properties to Cadabra objects inside a python file. Could someone help me with finding the right input syntax, for instance for the property Indices?

Consider the following cadabra notebook contents:

{ a, b, c, d }::Indices(spacetime, independent=True)

This assigns the Indices property to the a,b,c,d objects in a cadabra notebook. I would like to do this inside a python notebook. Here is my attempt:

import cadabra2 as cad
example_index = ['a', 'b', 'c', 'd']
cad.Indices(example_index_set, space=spacetime, independent=True)

This returns the error that example_index is not a Cadabra object, so I'm not understanding the syntax for cad.Indices.

Thanks in advance!

evaluate() returns RuntimeError when using the explicit value of an index

Thu, 04 Jan 2024 16:16:00 +0000

Hi cdb experts and a happy new year (hopefully).

Please consider the following code snippet:

d = 3
{k,l,m,n}::Indices(rotation, values={1..3}).
\epsilon{#}::EpsilonTensor.

ex := (K_{1})_{m n} e_{1}_{n} = \epsilon_{1 m n} e_{1}_{n};
evaluate(ex, $e_{1}_{1} = 1, e_{1}_{2} = 0, e_{1}_{3} = 0$, rhsonly=True);

I am trying to evaluate the simple matrix multiplication K_1 e_1 where K_1 is the antisymmetric matrix \epsilon_{1 m n} and e_1 is the unit vector [1,0,0].

I get the error RuntimeError: No Indices property known for indices in EpsilonTensor.

On the other hand, if instead of the explicit index _{1} I use a general index _{l} then the calculation works (but it's not what I wanted to calculate):

d = 3
{k,l,m,n}::Indices(rotation, values={1..3}).
\epsilon{#}::EpsilonTensor.

ex := (K_{l})_{m n} e_{l}_{n} = \epsilon_{l m n} e_{l}_{n};
evaluate(ex, $e_{1}_{1} = 1, e_{1}_{2} = 0, e_{1}_{3} = 0$, rhsonly=True);

Any advice? Thank you!

GPN

Simplification of sum of terms r a^m + s a^m to (r+s)a^m

Mon, 01 Jan 2024 16:12:07 +0000

I'm trying to solve simple operator algebra problems invovling the harmonic oscillator creation and annihilation operators. So I want to do things like a b^n -> n b^(n-1) + b^n a where [a,b] = 1

I've defined a set of rules that almost get me there, but the final result does not fully simplify and I don't know why. For example if I try to reduce a b^5 it produces 2 b^4 + 3 b^4 + b^5 a

I'm not sure what I'm doing wrong here. If there is a better way to go about doing this kind of simplification please let me know.

My code is as follows:

a::Symbol;
b::Symbol;

{a,b}::NonCommuting;
{m,n,r,s}::Integer.

r1 := {
a^0 -> 1,
b^0 -> 1,
b^1 -> b,
a b^m  -> (1+b a) b^{m-1},
a b -> 1 + b a,
b^m  b -> b^{m+1},
b  b^m -> b^{1+m}
};

def act(ex):
    converge(ex):
        distribute(ex)
        expand(ex)
        substitute(ex,r1)
        sort_sum(ex)
        collect_factors(ex)
        collect_terms(ex)
        simplify(ex)
    return ex

act($a b^5$);

Display "hand-made" latex

Fri, 29 Dec 2023 13:01:02 +0000

Hi experts.

In a Jupyter notebook with Cadabra2 kernel:

Is there any way to display "hand made" Latex, such as

s = Latex(
    '\\begin{array}{l} \n'
    ' 0 &  0 &  0 \\   \n'
    ' 0 &  0 &  1 \\   \n'
    ' 0 & -1 &  0      \n'
    '\\end{array}      \n'
)._repr_latex_()

Trying to display the above with

s;

prints it out as a string not as a properly formated array.

I am working in

Jupyter notebook
Cadabra2 kernel
ipynb file
Cadabra 2.4.5
Ubuntu 22.04

Thank you

GPN

Replacing expressions in a matrix

Wed, 27 Dec 2023 12:25:35 +0000

Seasons greetings cdb experts, and many thanks in advance!

In the code below I first create the set of matrices Kl, then try to pick out the components of one of the matrices (K1 in the example) and display them.

My actual goal is to display a matrix with the right values in a nice LaTeX format.

If you run the code below, you will notice that I try to change the i,j element of a matrix but despite the element.replace(), the final TheK1Matrix has values {12, 21} and not values {1, -1}. My questions are:

How to set the i,j element of the matrix below?

from cdb.core.component import *
from  cdb.utils.node import *

{k,l,m,n}::Indices(rotation, values={1,2,3}).
\epsilon{#}::EpsilonTensor.
Kl := (K_{l})_{m n} = \epsilon_{l m n};
evaluate(Kl, rhsonly=True);

TheK1Matrix := [[0,0,0],[0,0,12],[0,21,0],].
for ex in Kl[r'\equals']:
    lhs = get_lhs(ex.ex())
    rhs = get_rhs(ex.ex())
    if ${a?,b?,c?}$.matches(lhs):
        n = int(str(nth_arg(lhs,0)))
        if n == 1:
            i = int(str(nth_arg(lhs,1)))
            j = int(str(nth_arg(lhs,2)))
            for element in TheK1Matrix[i-1][j-1]:
                element;
                rhs;
                element.replace(rhs);
TheK1Matrix;

Ultimately, my goal is to display that matrix using "nice" LaTeX formatting, of the general form

\begin{array}
    0 & 0 & 0 \\
    0 & 0 & 1 \\
    0 & -1 & 0
\end{array}

How to achieve that?

Thank you

GPN

Displaying a matrix with matplotlib error "non-GUI backend"

Mon, 25 Dec 2023 11:07:25 +0000

Hi cdb experts.

I am in the

Jupyter notebook with Cadabra2 kernel
Ubuntu 22.04

When trying to use matplotlib, I get the error

UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.

and when I specifically set the backend to TkAgg I get

UserWarning: Matplotlib is currently using TkAgg, which is a non-GUI backend, so cannot show the figure

even though Tk is a GUI backend.

I have set the pythonpath to

PYTHONPATH=/usr/lib/python3.10/:/usr/lib/python3.10/dist-packages:/usr/lib/python3.10/tkinter/

and I do have both matplotlib and tkinter ("python3-tk") installed. What am I doing wrong, please?

Thank you!

P.S. As an example, the following code produces said error:

import matplotlib.pyplot as plt
from cdb.core.component import *

{i,j,k,l,m,n}::Indices(rotation, values={1,2,3}).
\epsilon{#}::EpsilonTensor.
Ki := (K_{i})_{m n} = \epsilon_{i m n}.
evaluate(Ki, rhsonly=True);

fig = plt.figure()

with plt.rc_context({'backend':'tkagg'}):
    for i in range(3):
        for j in range(3):
            val = get_component(Ki, $1, 0, 0$)
            plt.text(i,j,str(val), va='center',ha='center')

    plt.show()

Many thanks!

Displaying components of the epsilon matrix

Sun, 24 Dec 2023 11:56:20 +0000

Hi.

Two questions please.

Please consider the following code:

from cdb.core.component import *

{i,j,k,l,m,n}::Indices(topological,)
\epsilon::EpsilonTensor.
Ki := (K_{i})_{m n} = \epsilon_{i m n}.
get_component(Ki, $0, 0, 0$);

The answer it provides is entirely correct (\epsilon_{0 0 0}) but

I would like it to go the next step and display the actual value 1.
Is there any way to obtain all the components of the complete matrix Ki for a given i?

Of course, if I have the answer to Q1, I can do a for-loop and get the answer to Q2. Ideally, I would like some function that provides a list or dict which I can then further format.

Many thanks as always!

GPN

LaTeXForm expansion fails with "free indices do not match"

Fri, 22 Dec 2023 11:23:54 +0000

Hi and thanks as always for cdb.

Please consider the following definition of an "inner product":

{α, β}::Indices(spacetime).
σ::LaTeXForm("\mathbf{σ}").
σ::DifferentialForm(degree=1).

e_{idx?}::LaTeXForm("\mathbf{e}", "_{", idx?, "}").

inner{a?? b??}::LaTeXForm("\langle",a??,", ",b??,"\rangle").

This works nicely when I write

σβ := inner{σ e_{β}} ;

but it fails when I write

σβ := σ_{β} = inner{σ e_{β}} ;

The error is

RuntimeError: Free indices on lhs and rhs do not match

What am I doing wrong please?

Thank you

GPN

How to collect terms containing derivatives

Fri, 15 Dec 2023 14:55:16 +0000

Hello

I'm running the following:

{a#,b#}::Indices.
\delta{#}::KroneckerDelta.
{a#,b#}::Integer(0..3)
\partial{#}::PartialDerivative.
\epsilon{#}::EpsilonTensor(delta=-\delta).
h_{a1 b1}::Symmetric.
toh := {h_{a1 a1} = h, h_{a1}^{a1} = h}.
def LLmanip(ex):
    product_rule(ex) 
    distribute(ex)
    product_rule(ex)
    distribute(ex)
    epsilon_to_delta(ex)
    expand_delta(ex)
    distribute(ex)
    eliminate_kronecker(ex)
    canonicalise(ex)
    rename_dummies(ex)
    distribute(ex) 
    sort_product(ex)
    sort_sum(ex)
    substitute(ex, toh)
    lower_free_indices(ex)
    canonicalise(ex)
    rename_dummies(ex)
    collect_factors(ex)
    return(ex)

LL := \delta^{a1 b1 a2 b2 a3 b3 a4 b4} \partial_{a1}{h_{a2 b2}} \partial_{b1}{h_{a3 b3}} h_{a4 b4};
LLmanip(LL);

The result i get has these two terms:

\partial_{a1}{h_{a1 a2}} \partial_{a2}{h} h

and

\partial_{a1}{h} \partial_{a2}{h_{a1 a2}} h

which are the same but are not collected into one.

Is there some way to combine the terms?

Thanks

Partial derivative of a vector

Wed, 06 Dec 2023 14:04:22 +0000

Hello!

I am very a new Cadabra user and I am trying to expand partial derivative of a vector field A i.e. $\partial_{\mu} { A^{\mu} }$ with the following snippet of code:

{A^t,A^x, A^y, A^z}::Depends(\partial{#});
A := {A^{t}=A^t,A^{x}=A^x, A^{y}=A^y, A^{z}=A^z};
dA:=\partial_{\mu}{ A^{\mu} };
evaluate(dA,A);

which returns an error: RuntimeError: Dependencies on derivatives are not yet handled in the SymPy bridge

I ideally want it to give me the componenet expansion like: $ \partial_{\mu} { A^{\mu} } = - \partial_t A^t + \partial_x A^x + \partial_y A^y + \partial_z A^z $.

Also I want it to return a 2x2 matrix when I try to evaulate $ \partial_{\mu} { A^{\nu} } $.

Thanks in advance!

A vector-field basis of derivatives?

Mon, 04 Dec 2023 22:23:50 +0000

Hi Cdb experts. Thanks as always for your help.

I am trying to write down a vector field as is common in differential geometry, with a basis of derivatives. For example:

u := u^{\mu} \partial_{\mu};

My difficulty is that any serious calculation I try to do comes out all zeros. For example,

from cdb.relativity.abstract import *
u := u^{\mu} \partial_{\mu};
v := v^{\mu} \partial_{\mu};
ex := @(u)@(v) - @(v)@(u);

Is there any way to make cdb aware that should be considered on operator "waiting" for something to operate upon?

Thank you!

GPN

Avoiding index classes when handling ExNodes

Sat, 02 Dec 2023 03:58:14 +0000

I'm curious if other people have good strategies for avoiding index clashes when handling ExNodes directly.

I have in mind the example of the covariant derivative in https://cadabra.science/notebooks/ref_programming.html, which works fine so long as the index p on the connection doesn't appear in the original expression. (If it does, it breaks.) As far as I can tell, introducing indices when hacking ExNodes directly can easily lead to this.

Presumably the only solution is to work hard to avoid index clashes? This is what substitute does behind the scenes when it's running, I gather.