# Strings on a curved spacetime

1. Suppose we are interested in in string on a specific metric G, is it necessary to include a Dilaton field on back ground in order to preserve the Weyl invariance? suppose the spacetime is not empty, for example consider a cosmological constant term too.

2. In Polchinski's book it is shown that the Einstein's equation can be emerged from string theory as a consequence of the Weyl invariance but, it is true only to the first order of "Alpha-Prime". Does it mean that in the framework of string theory, Einstein Eq. is credited only in weak gravitational interactions?

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the following comment may not be directly relevent to the questions but, it seems funny so I decided to put it here The existence of the dilatonic fields on spcatime destroys the equivalence principle because of force from dilaton exchange. by giving mass to this field we can make its range finite (very very short actually), so the principle make sense in the usual scales !! –  Mohsen Feb 24 '13 at 15:32

1. The world sheet Weyl invariance requires/implies that the equations of motion for the spacetime fields are satisfied. These equations of motion constrain the dilaton (a field in the spacetime) as well. However, there exist solutions (i.e. also allowed CFTs) that have a constant dilaton and there also exist solutions that have a non-constant dilaton. Whether a particular background metric with a constant or a non-constant dilaton is a solution (i.e. defines a CFT without a Weyl anomaly) could only be decided if you said what background you are actually talking about.

2. The $\alpha'$ corrections become important in very strong gravitational fields and/or very high energies. In terms of curvature, for example, $\alpha' R^2$ becomes comparable to $R$ (Riemann or Ricci tensor) when $R\sim 1/\alpha'$ i.e. the curvature radius is comparable to $l_{\rm string}$, an ultramicroscopic distance scale. This has clearly no measurable impact on astronomical objects. However, it does become important when one studies the character of spacetime near the Planck/string length. Let me emphasize that general relativity implies a whole tower of nonlinear (relativistic) corrections to Newton's gravitational law and string theory reproduces this whole tower! The terms suppressed by $\alpha'$ are much smaller and smaller in a different sense than the nonlinear corrections resulting from GR are smaller than the leading terms in Newton's theory. The theory resulting from string theory obeys all the principles of general relativity – just to be sure. For example, large black holes (strong gravitational fields in GR) are described by the same equations in string theory as in "ordinary" general relativity; the corrections from string theory are negligible because the curvature radius is much longer than the string or Planck scale.

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Thank you for your answer. however I didn't asked the first question well. –  Mohsen Feb 24 '13 at 14:59
in the question 1, I was trying to say that if we choose a particular metric for spacetime and this socaetime curvature comes from a Cosmological Constant, but set the Kalb-Ramond field to zero, then is it necessary to have a Dilation Field on spcaetime? as result of weyl invariance. In order to be clear I mean the background of Randall Sundrum Model. –  Mohsen Feb 24 '13 at 15:07
Dear Mohsen, you use the words "particular metric" but you didn't tell us any particular metric so your question can't be answered. A profile of the dilaton field may sometimes be chosen so that the equations of motion are solved for a metric you choose. Moreover, you are including the Kalb-Ramond field. Its role here is exactly the same as the dilaton and the metric. All these fields enter equations of motion that must be solved if the CFT world sheet theory is free of the Weyl anomaly. –  Luboš Motl Feb 25 '13 at 5:53
On the other hand, the cosmological constant isn't a field - it's a constant - but it also enters the equations of motion (Einstein's equations). Concerning RS, a meaningful question similar to yours is whether RS may be embedded in string theory (if you were trying to ask this question, you failed and only revealed some confusion of yours about string theory itself). There are papers on that - with some disclaimers, RS may be embedded in string theory. –  Luboš Motl Feb 25 '13 at 5:55
My exact problem is (Actually my Master project) : I want to use a geometry like the one used in RS model, I mean a D-dimensional space-time with a metric similar to randal sundrum model and of course we have two D-branes. then at the first step, I want to quantize some kind of strings (for example open strings that are extended from one brane to the other one) and find their spectrum for example. But, in this comment I my question is, if I want to use this geometry as background and, of course,.... –  Mohsen Feb 26 '13 at 14:27