# The sign of the Tides tensor is incorrect

+1 vote

Hello, I'm doing the calculation of the Tide tensor, and although the expression is correct at the end, the sign of the expression is incorrect. That makes me think that the problem is in some configuration of the simmetries of the Riemann tensor. The problem might be a result of a poor use of the system by me. I'll put the code for Reissner-NordstrÃ¶m spacetimes as an example

{r,t,\phi,\theta}::Coordinate; {\mu,\nu,\rho,\sigma,\lambda,\kappa,\chi,\gamma, \tau, \beta, a, b}::Indices(values={t,r,\phi,\theta}, position=fixed); \partial{#}::PartialDerivative; g_{\mu\nu}::Metric. g^{\mu\nu}::InverseMetric. R^{\rho}_{\sigma\mu\nu}::RiemannTensor.

ss:= { g{t t} = 1-2*M/r + q2/r2,
g
{r r} = -1/(1-2*M/r + q2/r2), g{\theta\theta} = -r**2, g{\phi\phi}=-r2 \sin(\theta)2 }.

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);

rm:= R^{\rho}_{\sigma\mu\nu} = \partial_{\mu}{\Gamma^{\rho}_{\nu\sigma}} -\partial_{\nu}{\Gamma^{\rho}_{\mu\sigma}} +\Gamma^{\rho}_{\mu\lambda} \Gamma^{\lambda}_{\nu\sigma} -\Gamma^{\rho}_{\nu\lambda} \Gamma^{\lambda}_{\mu\sigma}; substitute(rm, ch) evaluate(rm, ss, rhsonly=True);

ex:=K^{a}_{b}=R^{\mu}_{\rho\nu\beta}ei^{a}_{\mu}i^{\rho}_{t}i^{\nu}_{b}i^{\beta}_{t}; substitute(ex, rm) viel:={i^{t}_{t}=Q/f,i^{r}_{t}=-j,i^{\theta}_{t}=0,i^{\phi}_{t}=0,i^{t}_{r}=-j/f,i^{r}_{r}=Q,i^{\theta}_{r}=0,i^{\phi}_{r}=0,i^{t}_{\theta}=0,i^{r}_{\theta}=0,i^{\theta}_{\theta}=1/r,i^{\phi}_{\theta}=0,i^{t}_{\phi}=0,i^{r}_{\phi}=0,i^{\theta}_{\phi}=0,i^{\phi}_{\phi}=1/(r*\sin(\theta)),ei^{t}_{t}=f*Q*/((Q**2-j**2)),ei^{r}_{t}=-f*j/((j**2-Q**2)),ei^{\theta}_{t}=0,ei^{\phi}_{t}=0,ei^{t}_{r}=-j/((j**2-Q**2)),ei^{r}_{r}=Q/((-f**2+Q**2)),ei^{\theta}_{r}=0,ei^{\phi}_{r}=0,ei^{t}_{\theta}=0,ei^{r}_{\theta}=0,ei^{\theta}_{\theta}=r,ei^{\phi}_{\theta}=0,ei^{t}_{\phi}=0,ei^{r}_{\phi}=0,ei^{\theta}_{\phi}=0,ei^{\phi}_{\phi}=r*\sin(\theta)}; substitute(viel, $j=(Q**2-f)**(1/2)$) substitute(viel, $f=1-2*M/r + q**2/r**2$) evaluate(ex, viel, rhsonly=True);

Hi Gabriel. I'd suggest that you improve the quality of your post.

First, the code is not well rendered, so if I copy from the post and run it, if fails! I spent about 20 minutes modifying it just to go through the evaluations.

And finally, it good be great if you share the obtained and expected results. By the way, it might be as simple as a confusion with the signature of a term, I mean the usual (+,-,-,-) vs (-,+,+,+) confrontation.

I'm sorry, I have to mention that I'm using the cadabra cloud, so codes on it don't really work on the desktop version I was told.

To explain the code, I'm using the Reissner-NordstrÃ¶m metric with the (-,+,+,+) signature. To calculate the tides tensor i'm using a vierbein, the first vierbein is chosen to be the velocity of a particle in-going radially.