Ohm’s law has a gravitational analog that relates energy flow (current) from a black hole, or during gravitational collapse, to the “gravitational electric” field (see Sec. 4.4 of [11]). Perhaps the instability of the black hole can be summarized in a negative “gravitational conductivity”. 

A potential application of our findings is to supercon- ducting cosmic strings, where massless QED in 1+1 di- mensions is expected to apply for fermion zero modes on the string [12]. Our analysis shows that if superconduct- ing strings really behave as 1+1 dimensional systems, they will carry oscillatory currents because of the back- reaction of the induced currents on the external electric fields. (Oscillatory currents were also discussed in [13], though these occurred due to the periodic dynamics of the strings.) 

While our analysis has enabled us to fully treat back- reaction of the Schwinger process, our results cannot be transported to 3+1 QED for two reasons. First, the elec- tron has a non-zero mass. For electric fields smaller than the electron mass squared, the Schwinger process is ex- ponentially suppressed and the vacuum is essentially an insulator. In situations where the electric field is larger than the electron mass squared, the exponential suppres- sion is absent and the dynamics may be closer to what we have found. The second reason is that the larger number of dimensions can change the picture dramatically. In 

[1] J. S. Schwinger, Phys. Rev. 82 , 664 (1951). 

[2] Y. Kluger, J. M. Eisenberg, B. Svetitsky, F. Cooper and E. Mottola, Phys. Rev. D 45 , 4659 (1992). Y. Kluger, J. M. Eisenberg, B. Svetitsky, F. Cooper and E. Mottola, Phys. Rev. Lett. 67 , 2427 (1991). F. Cooper and E. Mottola, Phys. Rev. D 40 , 456 (1989). 

[3] E. T. Akhmedov and P. Burda, arXiv:0912.3435 [hep-th]. 

[4] D. Bazeia, F. A. Brito, W. Freire and R. F. Ribeiro, Int. J. Mod. Phys. A 18 , 5627 (2003) [arXiv:hep-th/0210289]. 

[5] I.S. Gradshteyn and I.M. Ryzhik Table Of Integrals, Se- ries and Products , fourth edition, Academic Press, New York (1980). 

[6] P. M. Morse and H. Feshbach, Methods of Theoretical Physics , Part II, McGraw Hill (1953), § 12.3 

1+1 dimensions, the inter-charge potential is linear and electric charge is confined. In 3+1 dimensions, electric charges interact by the Coulomb potential and are not confined. This suggests that our system may be closer to the case of chromo-electric fields in 3+1 dimensions with massless quarks. While this has some features that re- semble the model we have considered, there are essential differences due to the non-Abelian nature of the interac- tions. 

We thank Edward Witten for suggesting the mass- less Schwinger model for the backreaction problem, Ioan- nis M. Besieris for sharing his notes on the solution to the massive Klein-Gordon equation, and Ratin Akhoury, Daniel Green, Zohar Komargodski, Juan Maldacena, Dmitry I. Podolsky, David Shih and Yuji Tachikawa for helpful comments and discussions. We thank Peter J. Kernan and Pascal M. Vaudrevange for invaluable com- puting help. This work was supported by the U.S. De- partment of Energy at Case Western Reserve Univer- sity. TV was also supported by grant number DE-FG02- 90ER40542 at the Institute for Advanced Study. 

[7] S. R. Coleman, R. Jackiw and L. Susskind, Annals Phys. 93 , 267 (1975); S. R. Coleman, Annals Phys. 101 , 239 (1976). 

[8] K. Fujikawa, arXiv:0902.2066 [hep-th]. 

[9] C. M. Naon, Phys. Rev. D 31 , 2035 (1985). 

[10] J. von Delft and H. Schoeller, Annalen Phys. 7 , 225 (1998) [arXiv:cond-mat/9805275]. 

[11] “General Relativity”, R.M. Wald, University of Chicago Press, Chicago (1984). 

[12] E. Witten, Nucl. Phys. B 249 , 557 (1985). 

[13] M. Aryal, A. Vilenkin and T. Vachaspati, Phys. Lett. B 194 , 25 (1987). 

## Acknowledgments 

