Lamb shift in Schwarzschild spacetime

Wenting Zhou & Hongwei Yu

Department of Physics, Hunan Normal University, Changsha, Hunan, China

OUTLINE

Why

How--DDC formalism

Lamb shift in Schwarzschild spacetime

Summary

What is Lamb shift?

Experimental discovery:

In 1947, Lamb and Retherford show that the level lies about 1000MHz, or 0.030cm-1 above the level . Then a more accurate value 1058MHz.

Theoretical result:

Dirac theory in Quantum Mechanics shows: the states, 2s1/2 and 2p1/2 of hydrogen atom are degenerate.

Why

The Lamb shift

Important meanings

Physical interpretation

The lamb shift results from the coupling of the atomic electron to the vacuum electromagnetic field which was ignored in Dirac theory.

Our interest

How spacetime curvature affects the Lamb shift? Observable?

DDC (J. Dalibard, J. Dupont-Roc and C. Cohen- Tannadji) formalism

J. Dalibard J. Dupont-Roc C. Cohen-Tannadji

Model: a two-level atom coupled with vacuum scalar field.

Atomic states: and , with energies . 2

0

Atom initial state , that of the field is .b 0

The Hamiltonian of atom-field system:

IFAsys HHHH

with

Integrationsf EEE

—— corresponding to the effect of vacuum fluctuationsfE—— corresponding to the effect of radiation reaction

sE

Heisenberg equation

of the field

Heisenberg equation

of the atom

The atomic dynamical equation of G

Physical interpretation of the evolution of the atomic observable.

Symmetric operator orderinguncertain?

Phys. Rev. A 50, 1755 (1994),

Phys. Rev. A 52, 629 (1995).

J. Phys. (Paris) 43, 1617 (1982);

J. Phys. (Paris) 45, 637 (1984);

The Lamb shift in Schwarzschild spacetime

22222122 sin)/21()/21( ddrdrrMdtrMds

A complete set of modes functions satisfying the Klein-Gordon equation:

outgoing

ingoing

spacetime gauge field modesKlein-Gordon equation

,0)|()(22

2

rRrVdr

dll

,2)1(2

1)(32

r

M

r

ll

r

MrVlwith ).12/ln(2* MrMrr

It is difficult to express the solution in terms of the elementary functions, but two classes of solutions in the asymptotic regions (V(r)~0) single out:

The field operators are expanded in terms of these basic modes, then we can define the vacuum state.

Three vacuum states:

Positive frequency modes → the Schwarzschild time t.

The positive frequency modes incoming from → the Schwarzschild time t,The positive frequency modes emanate from the past horizon → the Kruskal coordinate .

The positive frequency modes incoming from → the Kruskal coordinate The positive frequency modes emanate from the past horizon → the Kruskal coordinate .

1. Boulware vacuum

2. Unruh vacuum

3. Hartle-Hawking vacuum

It describes the state of a spherical massive body.

It describe the state of a black hole after the collapsing of a massive body.

It describe the state of a black hole in thermal equilibrium with thermal radiation.

How the atomic energy is shifted in such backgrounds?

Consider the Lamb shift of a static atom fixed in the exterior region of the spacetime with a distance r from the mass center.

a. The Lamb shift in Boulware vacuumThe revision is caused by

spacetime curvature.

The corresponding Lamb shift of a static one in Minkowski spacetime with no boundaries. It is logarithmically divergent as a result of non-relativistic treatment here and can be removed by introducing a cutoff factor.

The grey-body factor

Consider the geometrical approximation:

The effect of backscattering of field modes off the curved geometry.

3Mr

2M

Vl(r)

,max2 V ;1~lB

,max2 V .0~lB

2. Near r~3M, f(r)~1/4, the revision is 25%!

If so, it is observable if we have such a massive body!

However, the above result is valid only in the asymptotic regions, the shift of the atom at arbitrary position requires specifics about the radial functions that is not completely explicit so far.

1. In the asymptotic regions, i.e. and , f(r)~0, the revision is negligible!

Mr 2 r

Discussion:

b. The Lamb shift in Unruh vacuum

The corresponding temperature:

Plankian factor

TMr ,2As , the temperature the atom feels is divergent.

Physical interpretation: In order to keep a fixed distance from the mass center near the event horizon, the atomic acceleration relative to the freely-falling frame reference blows up, it is just the acceleration gives rise to the extra effect.

Compared with that in Boulware vacuum:

TMU rfMr )](1[,2when

We deduce that:

1. There is thermal radiation emanate from the black hole event horizon!

2. The field modes that emanate from the event horizon is backscattered by the spacetime curvature through its way to infinity.

3. The flux it partly depleted and weakened from the event horizon to infinity.

TMU rfrfr )()](1[,when

rTrB rf ])([|

rB |

M

Low temperature limit

High temperature limit

For ,T

It is always finite, especially, in the following two limited cases:

Phys. Rev. D 82, 104030 (2010);

c. The Lamb shift in Hartle-Hawking vacuum

1. There is thermal radiation at infinity in Hartle-Hawking vacuum, and the corresponding temperature is the usual Hawking temperature, i.e.,

2. The result reveals in another context that the Hartle-Hawking vacuum describes a state of a black hole in equilibrium with black-body radiation at infinity.

,rwhen

We deduce that:

Summary

As opposed to the atomic Lamb shift in Minkowski spacetime, the spacetime curvature affects the atomic Lamb shift.