# Boulware, Deser ## String-Generated Gravity Models \[Links: [DOI](https://doi.org/10.1103/PhysRevLett.55.2656)\] \[Abstract: Expansion of supersymmetric string theory suggests that the leading quadratic curvature correction to the Einstein action is the [[0425 Gauss-Bonnet gravity|Gauss-Bonnet]] invariant. We show that this model has both flat and anti-de Sitter space as solutions, but that the cosmological branch is unstable, because the graviton becomes a ghost there: The theory solves its own cosmological problem. The general static spherically symmetric solution is exhibited; it is asymptotically Schwarzschild. The sign of the Gauss-Bonnet coefficient determines whether there is a normal event horizon (for the string-generated sign) or a naked singularity. We discuss the effects of [[0006 Higher-derivative gravity|higher-curvature corrections]] and of an explicit cosmological term on stability.\] ## Summary - one of the original papers on finding the spherically symmetric solution in GB gravity (independently by [[1986#Wheeler]]) # Dray, t'Hooft ## The gravitational shock wave of a massless particle \[Links: [Inspire](https://inspirehep.net/literature/203743), [DOI](https://doi.org/10.1016/0550-3213(85)90525-5)\] \[Abstract: The (spherical) gravitational [[0117 Shockwave|shock wave]] due to a massless particle moving at the speed of light along the horizon of the Schwarzchild black hole is obtained. Special cases of our procedure yield previous results by [[1971#Aichelburg, Sexl|Aichelburg and Sexl]] for a photon in Minkowski space and by Penrose [2] for sourceless shock waves in Minkowski space. A new derivation of the (plane) shock wave of a photon in Minkowski space [1] involving explicit calculation of geodesics crossing the shock wave is also given in order to clarify the underlying physics. Applications to quantum gravity, specifically the possible effect on the Hawking temperature, are briefly discussed.\] ## Summary - spherical shock wave solution due to a ==massless== particle moving ==along horizon== - a new derivation of shock wave by a photon in Minkowski space found earlier by [[1971#Aichelburg, Sexl]] - using a gluing technique, i.e., junction conditions - with explicit calculation of geodesics crossing the shock wave ## Necessary and sufficient condition for describing a shock wave via a coordinate shift - ansatz $\mathrm{d} \hat{s}^{2}=2 A(u, v) \mathrm{d} u \mathrm{d} v+g(u, v) h_{i j}(x^{i}) \mathrm{d} x^{i} \mathrm{d} x^{j}$ - condition: $A_{,v}=0=g_{,e v}$ and $\frac{A}{g} \Delta f-\frac{g_{, u v}}{g} f=32 \pi p A^{2} \delta(\rho)$ - $\Delta$ is the Laplacian w.r.t. $h_{ij}$ - $f=f(x^i)$ is the shift in $v$ ## Continuous v.s. discontinuous coordinates - continuous coordinates - some metric satisfying vacuum Einstein equations$\mathrm{d} \hat{s}^{2}=2 A(u, v) \mathrm{d} u \mathrm{d} v+g(u, v) h_{i j}(x^{i}) \mathrm{d} x^{i} \mathrm{d} x^{j}$ - introduce a shock wave by keeping the above solution for $u<0$ and replacing $v$ by $v+f(x^i)$ for $v>0$. $\mathrm{d} s^{2}=2 A(u, v+\theta f) \mathrm{d} u\left(\mathrm{d} v+\theta f_{, i} \mathrm{d} x^{i}\right)+g(u, v+\theta f) h_{i j} \mathrm{d} x^{i} \mathrm{d} x^{j}$ - discontinuous coordinates - $\hat{u}=u, \hat{v}=v+\theta f, \hat{x}^i=x^i$ - $\mathrm{d} s^{2}=2 A(\hat{u}, \hat{v}) \mathrm{d} \hat{u}(\mathrm{d} \hat{v}-\delta(\hat{u}) f \mathrm{d} \hat{u})+g(\hat{u}, \hat{v}) h_{i j} \mathrm{d} \hat{x}^{i} \mathrm{d} \hat{x}^{j}$ - note that the metric is in fact *continuous*, i.e. can go to a third coordinate system such that coefficients are all continuous, but that form is not very useful # Fradkin, Tseytlin (NPB261) ## Quantum String Theory Effective Action \[Links: [Inspire](https://inspirehep.net/literature/222174), [DOI](https://doi.org/10.1016/0550-3213(85)90559-0), [DOI-erratum](https://doi.org/10.1016/0550-3213(86)90522-5)\] \[Abstract: We present a covariant background field method for quantum string dynamics. It is based on the effective action $\Gamma$ for fields corresponding to different string modes. A formalism is developed for the calculation of $\Gamma$ in the $\alpha' \to 0$ limit. It is shown that in the case of closed Bose strings $\Gamma$ contains the standard kinetic terms for the scalar, external metric and the antisymmetric tensor. Our approach makes possible a consistent formulation and solution of a ground state problem (including the problem of space-time compactification) in the string theory. We suggest a solution to the old “tachyon problem” based on the generation of non-trivial vacuum values for the scalar field, metric and antisymmetric tensor. It is shown that a preferred compactification in the closed Bose string theory is to three (anti-de Sitter) space-time dimensions.\] # Fradkin, Tseytlin (PRB158) ## Effective Field Theory from Quantized Strings \[Links: [Inspire](https://inspirehep.net/literature/209126), [PRB](https://doi.org/10.1016/0370-2693(85)91190-6)\] \[Abstract: We present a covariant definition of the effective action $\Gamma$ for fields corresponding to different string excitations (scalar, metric, antisymmetric tensor, …) in terms of a string path integral with “external sources”. The problem of deriving the gravitational part of $\Gamma$ is reduced to that of establishing the effective action for a generalized $\sigma$-model on a curved two-dimensional background and subsequent integration over two-metrics. The low-energy approximation for $\Gamma$ is computed.\] # Fradkin, Tseytlin (PRB160) ## Effective Action Approach to Superstring Theory \[Links: [Inspire](https://inspirehep.net/literature/214088), [DOI](https://doi.org/10.1016/0370-2693(85)91468-6)\] \[Abstract: We define the quantum effective action for local fields corresponding to superstring excitation modes. It is given by a superstring theory path integral with a generalized superstring action containing couplings to background fields of $D = 10$ super-Yang-Mills and supergravity multiplets. Dilaton couplings in a low-energy approximation for the effective action are shown to be the same as in corresponding $D = 10$ supergravity actions.\] ## Refs - [[0505 Off-shell strings]] # Fukuyama, Kamimura ## Gauge Theory of Two-dimensional Gravity \[Links: [Inspire](https://inspirehep.net/literature/16530)\] \[Abstract: The theory of gravity in two dimensions is discussed as a $O(2,1)$ [[0557 BF theory|gauge theory]]. The pure gauge system gives a geometrical equation in which the Riemannian curvature is a constant. We obtain in conformal gauge the generators of the conformal algebra in terms of a [[0311 Topologically non-trivial black holes|Lionville]] field and an associated scalar. These appears no central charge in the classical algebra.\] # Jackiw ## Lower Dimensional Gravity \[Links: [Inspire](https://inspirehep.net/literature/204694)\] \[Abstract: Gravity theory on a line and in the plane is reviewed. The triviality of the planar Einstein model is avoided by adding sources and a topological mass term. A constant curvature model for two dimensional space-time, analogous to the theory in three dimensional space-time, is proposed.\]