skip to content

Department of Applied Mathematics and Theoretical Physics


Classical and quantum gravity.   This includes (i.) the study of black holes and their quantum radiation; (ii.) gravitational and other types of radiation emitted in ultra-relativistic collisions or close encounters; (iii.) interactions between massive bodies; (iv.) classical and quantum cosmology, particularly in the case of theories which generalise Einstein's theory so as to include local supersymmetry between bosons (force-carrying particles) and fermions (matter particles).  The most general such field theory of interest is known as gauge-invariant supergravity.

Under (i.) above, Andrew Farley (Ph.D. student 1997; Ph.D. approved 2002) and I were the first to show that quantum effects following gravitational collapse to a black hole are governed by quantum amplitudes, not just by probabilities, as had been posited since quantum radiation from black holes was first found in 1974/5.   Thus, there is no extra difficulty in treating gravitational fields according to the laws of quantum mechanics, beyond the usual conceptual difficulties already present in quantum theory applied to simpler non-gravitational systems.   In particular, the loss of phase posited since 1975 does not occur -- there is no difficulty with quantum information.

Under (ii.), calculations of classical gravitational radiation emitted in collisions or close encounters of two bodies, with relative speed close to the speed of light, have found an application (unexpected to the author) in string theory since 2001/2.   Such gravitational fields were originally studied in 1975--8, and subsequently with Philip Payne, as the only known examples of localised vacuum gravitational fields in which strong-field gravitational radiation can be found analytically.  More recently, they have provided a way of verifying or falsifying string-theory models experimentally (possibly in dimensions higher than four) in extremely high-energy particle collisions !

The work under (iii.) involves the mathematical technique of matched asymptotic expansions, to describe the motion of several black holes or other self-gravitating bodies, interacting by gravitational (and/or Yang-Mills) forces, together with the gravitational and other radiation emitted.   It provides the basis for a systematic treatment of gravitational interactions between well-separated bodies; the approach is summarised in the 1996 book on Black Holes (see below).

Under (iv.), consider, for example, quantum amplitudes involving localised (asymptotically-flat) fields, for a locally-supersymmetric field theory such as N=1 supergravity, in four space-time dimensions.   In Dirac's approach to theories with local symmetries, one naturally poses initial and final data (not the same as S-matrix or particle-scattering data) on a pair of space-like hypersurfaces separated by a time T.   Following Dirac's approach, one finds that the amplitude is semi-classical -- that is -- of the form  exp(-I) times known factors.   Here, I is the 'Euclidean action' of the classical solution which joins the initial to the final data, and  exp( ) denotes the exponential function. 

Quantum amplitudes in the more physically-realistic gauge-invariant N=1 supergravity models are also semi-classical, in this sense.   In all these models, one is 'thrown back' onto the study of classical solutions.   The gauge-invariant models incorporate the familiar symmetry groups, SU(2), SU(3),..., of experimental particle physics.   Are there particle-like classical bosonic/fermionic solutions for these gauge-invariant models?   If so, then their interactions (classical or quantum) can be studied via the matched-asymptotic approach of (iii.) above.   Also, do these gauge-invariant theories have simple, physically-interesting cosmological solutions, which include matter as well as gravity?

Selected Publications

  • D'Eath, P.D. 1996 'Supersymmetric Quantum Cosmology', 252 pages, in the series Cambridge Monographs on Mathematical Physics, (Cambridge University Press).
  • D'Eath, P.D. 1996 'Black Holes: Gravitational Interactions', 286 pages, in the series Oxford Mathematical Monographs, (Oxford University Press).
  • D'Eath, P.D. 2003 'What local supersymmetry can do for quantum cosmology'. In The Future of Theoretical Physics and Cosmology, ed. G.W.Gibbons, E.P.S.Shellard & S.J.Rankin (Cambridge University Press), 693-722. (arXiv gr-qc/0511042)
  • Farley, A.N.St.J. & D'Eath, P.D. 2004 'Scalar-field amplitudes in black-hole evaporation'. Phys. Lett. B 601, 184-191. (arXiv gr-qc/0407086)
  • Farley, A.N.St.J. & D'Eath, P.D. 2005 'Bogoliubov transformations for amplitudes in black-hole evaporation'. Phys. Lett. B 613, 181-188. (arXiv gr-qc/0510027)
  • Farley, A.N.St.J. & D'Eath, P.D. 2005 'Spin-1/2 amplitudes in black-hole evaporation'. Class. Quantum Grav. 22, 3001-3014. (arXiv gr-qc/0510036)
  • Farley, A.N.St.J. & D'Eath, P.D. 2006 'Quantum amplitudes in black-hole evaporation: Spins 1 and 2'. Annals of Physics (N.Y.) 321, 1334-1374. (arXiv gr-qc/0708.2013)
  • Farley, A.N.St.J. & D'Eath, P.D. 2007 'Quantum amplitudes in black-hole evaporation: coherent and squeezed states'. Class. Quantum Grav. 24, 105-127. (arXiv gr-qc/0708.2018)
  • D'Eath, P.D. & Farley, A.N.St.J. 2015 'Quantum amplitudes in black-hole evaporation with local supersymmetry'. In Quantum Aspects of Black Holes, ed. X.Calmet (Springer), 195-228.

Research Group

Relativity and Gravitation