# String Theory, understanding Einstein's Field Equations

Let me give some insight. I was researching about how Einstein-Field Equations arise from String Theory, and I came upon this post. It explains, mathematically, how you can arrive at the corrected Einstein Field Equations from string theory:

$${{R_{\mu \nu }} + 2{\nabla _\mu }{\nabla _\nu }\Phi - \frac14{H_{\mu\lambda \kappa }}H_\nu{}^{\lambda \kappa }} = 0$$

As pointed out in this answer of the previous questions.

The thing is, I would like to know what $$R_{\mu \nu}$$, $$\nabla_\mu$$, $$\nabla_\nu$$, $$\Phi$$, $$H_{\mu\lambda \kappa}$$ and $$H_{\nu}{}^{\lambda \kappa}$$ mean or represent in the real world and also, what the equation means as a whole. On the other hand, what does it have to do with the "original" Einstein's Field Equations?

$$R_{\mu \nu} - \frac{1}{2}g_{\mu \nu}R \;+ g_{\mu \nu} \Lambda = \frac{8 \pi G}{c^4}T_{\mu \nu}$$

The "string theory equations", have very little in common with the "original equations", the only thing I can identify is the Ricci Tensor $$R_{\mu \nu}$$, not even the metric tensor. Is there something analogous to the Cosmological Constant in the "string theory equations"?

Are these equations experimentally tested, for example, giving as solutions the metrics for different gravitational fields or how much light would bend around a gravitational field? Do these equations even predict something new?

Sidenote: I would like to understand the concepts from a didactic and divulgative way, because I'm not a professional, nor even have a degree in physics, but I would not disregard a mathematical explanation; I think I would not get lost on the explanation.

Also, is very probable that I have made some dumb mistake (I'm very much a beginner to this), so correct me if I'm wrong.

The two fields $$\Phi$$ and $$H_{\mu\nu\lambda}$$ correspond to string dilaton and the B-field, or Kalb-Ramond field ($$H_{\mu\nu\lambda}$$ is just its field strength). In their absence we have $$R_{\mu\nu}=0$$, which is Einstein Field Equations (EFE) in vacuum. It can be seen if you set stress-energy tensor in full EFE to zero and contract $$R_{\mu\nu}-g_{\mu\nu}R/2$$ with $$g_{\mu\nu}$$. Then both $$R$$ and $$R_{\mu\nu}$$ vanish. As for the dilaton and the B-field, think of them as coming from stress-energy tensor. There is no cosmological constant (CC) as you noticed, but this is in 26 dimensions for bosonic string, or in 10 dimensions for superstring. When you try to compactify to our four spacetime dimensions, you realize that there is practically infinite number of ways to do so (string landscape), different compactifications can give different cosmological constants in four dimensions. The sign of CC is whole another story.

• What about the experiments? Are these equations proved and agree with General Relativity? Commented Aug 6, 2021 at 10:11
• @ÁlvaroRodrigo these equations coincide with GR in 26 (or 10) dimensions in the presence of dilaton and B-field. After appropriate compactification you will have many different fields in addition to those. Some of these fields could give rise to dark matter and dark energy, but the details depend on how you compactify extra dimensions, so we cannot say anything definitively. Direct tests of string theory would require very high energies which are unavailable to us, but there are possible indirect effects of strings and people are working on them too.
– Kosm
Commented Aug 6, 2021 at 10:33
• So, in conclusion. If we were able to prove that String Theory is right, and also find the compactification that our own universe has, these equations would be right and agree with GR? Commented Aug 6, 2021 at 10:36
• @ÁlvaroRodrigo GR is not a problem, string theory always gives rise to GR at low energies. The problem is everything else: particle spectrum, Standard Model parameters, stabilisation of the vacuum, how to obtain inflation, dark energy and so on
– Kosm
Commented Aug 6, 2021 at 10:41
• @ÁlvaroRodrigo you mean popular science or technical material? For technical there are well-known books by Polchinski, by Becker-Becker-Schwarz, and by Michael Dine for example. Not familiar with pop science books, you can search in google or even here at stackexchange for recommendations.
– Kosm
Commented Aug 6, 2021 at 15:47