Memory Effects and the Infrared Triangle
By Ana Raclariu
In February 2016, the LIGO collaboration announced the first observation of gravitational
waves from a pair of inspiraling and merging black holes.1 The signal perfectly
matched the prediction of Einstein’s theory of general relativity from 100 years ago, which revolutionized our understanding of physics. The general theory of relativity is based on
the idea that matter or energy are sources of spacetime curvature, and particles follow paths
Along this curved geometry. More generally, if two massive objects come
close enough, they start orbiting each other. This ‘binary’ system of involving two objects is a source of gravitational waves, ripples in spacetime carrying away energy at the speed of light. The more massive the binary components and the closer and faster they’re orbiting each other, the greater the amplitude of the waves. But even for fast orbiting, massive objects such as black holes or neutron stars, this amplitude is extremely small: At LIGO, gravitational waves induce relative changes in the arms of the interferometers comparable to a change in the radius of
the earth by the size of a proton!
In addition to the oscillations in spacetime, gravitational radiation is also known to generate a permanent shift in spacetime.2 If the arms of the interferometer have the same lengths before the detection of a gravitational wave pulse, they will end up with a permanent relative displacement after the signal has passed. This is a small fraction of the strain generated by a gravitational wave and goes by the name gravitational memory. That is, this displacement carries information about the net gravitational flux detected over some time interval and has the
potential of being measured in the near future.3 While gravitational memory has been
known for 50 years, it was only about 5 years ago that memory effects were shown not to be
unique to gravity. In fact, they are universal phenomena arising in any system with massless
particles. For example, quantum electrodynamics (QED) describes photons (electromagnetic
radiation), electrons and their interactions, and quantum chromodynamics (QCD) describes the dynamics of quarks and gluons, the fundamental constituents of nuclei; despite their differences, both theories have memory effects.
In an ongoing project in Professor Andrew Strominger’s group at Harvard, we are studying the memory effect in QCD and how to measure it in heavy ion collision experiments. To understand how these collision experiments work, imagine scattering an electron off a proton. An electron of increasing energy can distinguish more of the internal structure of the proton, acting very much like a microscope of increasing resolution. Probing a proton at very high energies reveals that it consists of a collection of fundamental particles called quarks and gluons, each carrying some fraction of the proton’s energy. A picture emerges where one can describe the dynamics of these constituent particles in terms of the most energetic quarks radiating relatively lower energy gluons. Configurations with any number of such gluons are nearly indistinguishable from the vacuum—a state with no gluons at all—and result in an infinite vacuum degeneracy.
More broadly, my research explores some of the consequences (6-9) of the recently discovered equivalence (10-12) between memory effects, symmetries of asymptotically flat spacetimes (AFS) and soft theorems. A spacetime is asymptotically flat if its curvature vanishes
at large distances from some region. Since the typical distance between astrophysical objects
in our universe is much larger than their individual diameters, the exterior of any massive object can be approximated by an AFS. In the 1960s, Bondi, van der Burg, Metzner and Sachs (BMS) found that such geometries have an infinite dimensional asymptotic symmetry group.(13,14)
In contrast, flat spacetime only has a finite number of symmetries: translations, rotations and “boosts” (symmetries relating frames of reference moving at constant velocities with respect to one another). The infinite symmetry enhancement is due to so-called supertranslations, angle-dependent translations along light rays piercing a sphere surroundingsome region of spacetime.
Observable measurements such as scattering amplitudes in gravity in AFS must follow this symmetry. Together with the gravitational memory effect described before, the BMS symmetry forms a triangle of equivalences governing the low energy structure of gravity in AFS as shown in the figure below. Moreover, soft theorems are universal results governing the behavior of scattering amplitudes at zero or low energy which suggests that the low energy dynamics of many physical theories could be constrained by an infrared triangle.
The implications of the infrared triangle are far-reaching and have yet to be fully unraveled. Starting with the soft theorem corner of the triangle for some particular theory, one could discover new symmetries and gain a better understanding of its vacuum structure through the memory effect. All these ideas also hint toward a possible holographic description of gravity in flat space in terms of quantum theory in lower dimensions, a finding that could ultimately give us some insight into why information disappears into black holes.(15-18)
In February 2016, the LIGO collaboration announced the first observation of gravitational
waves from a pair of inspiraling and merging black holes.1 The signal perfectly
matched the prediction of Einstein’s theory of general relativity from 100 years ago, which revolutionized our understanding of physics. The general theory of relativity is based on
the idea that matter or energy are sources of spacetime curvature, and particles follow paths
Along this curved geometry. More generally, if two massive objects come
close enough, they start orbiting each other. This ‘binary’ system of involving two objects is a source of gravitational waves, ripples in spacetime carrying away energy at the speed of light. The more massive the binary components and the closer and faster they’re orbiting each other, the greater the amplitude of the waves. But even for fast orbiting, massive objects such as black holes or neutron stars, this amplitude is extremely small: At LIGO, gravitational waves induce relative changes in the arms of the interferometers comparable to a change in the radius of
the earth by the size of a proton!
In addition to the oscillations in spacetime, gravitational radiation is also known to generate a permanent shift in spacetime.2 If the arms of the interferometer have the same lengths before the detection of a gravitational wave pulse, they will end up with a permanent relative displacement after the signal has passed. This is a small fraction of the strain generated by a gravitational wave and goes by the name gravitational memory. That is, this displacement carries information about the net gravitational flux detected over some time interval and has the
potential of being measured in the near future.3 While gravitational memory has been
known for 50 years, it was only about 5 years ago that memory effects were shown not to be
unique to gravity. In fact, they are universal phenomena arising in any system with massless
particles. For example, quantum electrodynamics (QED) describes photons (electromagnetic
radiation), electrons and their interactions, and quantum chromodynamics (QCD) describes the dynamics of quarks and gluons, the fundamental constituents of nuclei; despite their differences, both theories have memory effects.
In an ongoing project in Professor Andrew Strominger’s group at Harvard, we are studying the memory effect in QCD and how to measure it in heavy ion collision experiments. To understand how these collision experiments work, imagine scattering an electron off a proton. An electron of increasing energy can distinguish more of the internal structure of the proton, acting very much like a microscope of increasing resolution. Probing a proton at very high energies reveals that it consists of a collection of fundamental particles called quarks and gluons, each carrying some fraction of the proton’s energy. A picture emerges where one can describe the dynamics of these constituent particles in terms of the most energetic quarks radiating relatively lower energy gluons. Configurations with any number of such gluons are nearly indistinguishable from the vacuum—a state with no gluons at all—and result in an infinite vacuum degeneracy.
More broadly, my research explores some of the consequences (6-9) of the recently discovered equivalence (10-12) between memory effects, symmetries of asymptotically flat spacetimes (AFS) and soft theorems. A spacetime is asymptotically flat if its curvature vanishes
at large distances from some region. Since the typical distance between astrophysical objects
in our universe is much larger than their individual diameters, the exterior of any massive object can be approximated by an AFS. In the 1960s, Bondi, van der Burg, Metzner and Sachs (BMS) found that such geometries have an infinite dimensional asymptotic symmetry group.(13,14)
In contrast, flat spacetime only has a finite number of symmetries: translations, rotations and “boosts” (symmetries relating frames of reference moving at constant velocities with respect to one another). The infinite symmetry enhancement is due to so-called supertranslations, angle-dependent translations along light rays piercing a sphere surroundingsome region of spacetime.
Observable measurements such as scattering amplitudes in gravity in AFS must follow this symmetry. Together with the gravitational memory effect described before, the BMS symmetry forms a triangle of equivalences governing the low energy structure of gravity in AFS as shown in the figure below. Moreover, soft theorems are universal results governing the behavior of scattering amplitudes at zero or low energy which suggests that the low energy dynamics of many physical theories could be constrained by an infrared triangle.
The implications of the infrared triangle are far-reaching and have yet to be fully unraveled. Starting with the soft theorem corner of the triangle for some particular theory, one could discover new symmetries and gain a better understanding of its vacuum structure through the memory effect. All these ideas also hint toward a possible holographic description of gravity in flat space in terms of quantum theory in lower dimensions, a finding that could ultimately give us some insight into why information disappears into black holes.(15-18)
Works Cited
[1] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116, no. 6, 061102 (2016) [arXiv:1602.03837 [gr-qc]].
[2] Zel’dovich, Y. B. and Polnarev, A. G., “Radiation of gravitational waves by a cluster of superdense stars”, Sov. Astron. AJ (Engl. Transl.), v. 18, no. 1 (Jul, 1974).
[3] P. D. Lasky, E. Thrane, Y. Levin, J. Blackman and Y. Chen, “Detecting gravitational wave memory with LIGO: implications of GW150914,” Phys. Rev. Lett. 117, no. 6,
061102 (2016) [arXiv:1605.01415 [astro-ph.HE]].
[4] A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,”arXiv:1703.05448 [hep-th].
[5] M. Pate, A. M. Raclariu and A. Strominger, “Color Memory: A Yang-Mills Analog of Gravitational Wave Memory,” Phys. Rev. Lett. 119, no. 26, 261602 (2017)
[arXiv:1707.08016 [hep-th]].
[6] D. Kapec, A. M. Raclariu and A. Strominger, “Area, Entanglement Entropy and Supertranslations at Null Infinity,” Class. Quant. Grav. 34, no. 16, 165007 (2017)
[arXiv:1603.07706 [hep-th]].
[7] D. Kapec, P. Mitra, A. M. Raclariu and A. Strominger, “2D Stress Tensor for 4D Gravity,” Phys. Rev. Lett. 119, no. 12, 121601 (2017) [arXiv:1609.00282 [hep-th]].
[8] T. He, D. Kapec, A. M. Raclariu and A. Strominger, “Loop-Corrected Virasoro Symmetry of 4D Quantum Gravity,” JHEP 1708, 050 (2017) [arXiv:1701.00496 [hep-th]].
[9] D. Kapec, M. Perry, A. M. Raclariu and A. Strominger, “Infrared Divergences in QED, Revisited,” Phys. Rev. D 96, no. 8, 085002 (2017) [arXiv:1705.04311 [hep-th]].
[10] A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 1407, 152 (2014) [arXiv:1312.2229 [hep-th]].
[11] T. He, V. Lysov, P. Mitra and A. Strominger, “BMS supertranslations and Weinbergs soft graviton theorem,” JHEP 1505, 151 (2015) [arXiv:1401.7026 [hep-th]].
[12] A. Strominger and A. Zhiboedov, “Gravitational Memory, BMS Supertranslations and Soft Theorems,” JHEP 1601, 086 (2016) [arXiv:1411.5745 [hep-th]].
[13] H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, “Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems,” Proc. Roy. Soc. Lond. A 269, 21 (1962).
[14] R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times,” Proc. Roy. Soc. Lond. A 270, 103 (1962).
[15] S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. 140, B516 (1965).
[16] S. W. Hawking, M. J. Perry and A. Strominger, “Soft Hair on Black Holes,” Phys. Rev. Lett. 116, no. 23, 231301 (2016) [arXiv:1601.00921 [hep-th]].
[17] S. W. Hawking, M. J. Perry and A. Strominger, “Superrotation Charge and Supertranslation Hair on Black Holes,” JHEP 1705, 161 (2017) [arXiv:1611.09175 [hep-th]].
[18] A. Strominger, “Black Hole Information Revisited,” arXiv:1706.07143 [hep-th].
[1] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116, no. 6, 061102 (2016) [arXiv:1602.03837 [gr-qc]].
[2] Zel’dovich, Y. B. and Polnarev, A. G., “Radiation of gravitational waves by a cluster of superdense stars”, Sov. Astron. AJ (Engl. Transl.), v. 18, no. 1 (Jul, 1974).
[3] P. D. Lasky, E. Thrane, Y. Levin, J. Blackman and Y. Chen, “Detecting gravitational wave memory with LIGO: implications of GW150914,” Phys. Rev. Lett. 117, no. 6,
061102 (2016) [arXiv:1605.01415 [astro-ph.HE]].
[4] A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,”arXiv:1703.05448 [hep-th].
[5] M. Pate, A. M. Raclariu and A. Strominger, “Color Memory: A Yang-Mills Analog of Gravitational Wave Memory,” Phys. Rev. Lett. 119, no. 26, 261602 (2017)
[arXiv:1707.08016 [hep-th]].
[6] D. Kapec, A. M. Raclariu and A. Strominger, “Area, Entanglement Entropy and Supertranslations at Null Infinity,” Class. Quant. Grav. 34, no. 16, 165007 (2017)
[arXiv:1603.07706 [hep-th]].
[7] D. Kapec, P. Mitra, A. M. Raclariu and A. Strominger, “2D Stress Tensor for 4D Gravity,” Phys. Rev. Lett. 119, no. 12, 121601 (2017) [arXiv:1609.00282 [hep-th]].
[8] T. He, D. Kapec, A. M. Raclariu and A. Strominger, “Loop-Corrected Virasoro Symmetry of 4D Quantum Gravity,” JHEP 1708, 050 (2017) [arXiv:1701.00496 [hep-th]].
[9] D. Kapec, M. Perry, A. M. Raclariu and A. Strominger, “Infrared Divergences in QED, Revisited,” Phys. Rev. D 96, no. 8, 085002 (2017) [arXiv:1705.04311 [hep-th]].
[10] A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 1407, 152 (2014) [arXiv:1312.2229 [hep-th]].
[11] T. He, V. Lysov, P. Mitra and A. Strominger, “BMS supertranslations and Weinbergs soft graviton theorem,” JHEP 1505, 151 (2015) [arXiv:1401.7026 [hep-th]].
[12] A. Strominger and A. Zhiboedov, “Gravitational Memory, BMS Supertranslations and Soft Theorems,” JHEP 1601, 086 (2016) [arXiv:1411.5745 [hep-th]].
[13] H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, “Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems,” Proc. Roy. Soc. Lond. A 269, 21 (1962).
[14] R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times,” Proc. Roy. Soc. Lond. A 270, 103 (1962).
[15] S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. 140, B516 (1965).
[16] S. W. Hawking, M. J. Perry and A. Strominger, “Soft Hair on Black Holes,” Phys. Rev. Lett. 116, no. 23, 231301 (2016) [arXiv:1601.00921 [hep-th]].
[17] S. W. Hawking, M. J. Perry and A. Strominger, “Superrotation Charge and Supertranslation Hair on Black Holes,” JHEP 1705, 161 (2017) [arXiv:1611.09175 [hep-th]].
[18] A. Strominger, “Black Hole Information Revisited,” arXiv:1706.07143 [hep-th].
