Volume 71 B, number 1 PHYSICS LETTERS 7 November 1977

A RELATION BETWEEN THE E I N S T E I N A N D T H E Y A N G - M I L L S F I E L D E Q U A T I O N S

P. OLESEN The Niels Bohr Institute, University of Copenhagen, DK-2 I O0 Copenhagen ~, Denmark

Received 10 August 1977

We show that any solution of the vacuum Einstein equations is a double self dual solution of the 0(4) Yang-Mills equations.

In this note we shall point out that there exists a simple relation between the Einstein and the Yang- Mills field equations. It turns out that any solution of the vacuum Einstein equations is a double self dual so- lution of the curved space 0(4)Yang-Mills equations. Presumably the opposite is not true, and hence the Yang-MiUs field equations have a richer structure than the Einstein equations (contrary to what is often stated).

We start from the identity [1 ]

Ra~uv - R ~ u v - Baug~v - Bavg~u B~vgau -- Bf3ugav,

where ( l )

R**a~uv_ i [det _ "4 (_g)] -1 eaOpo eUv6h Rposh, (2)

and

1 BUL, = Ruv - Z gu, R" (3)

The double star curvature tensor is thus the double dual of the curvature tensor.

Imposing the vacuum Einstein equation

Ruv = O, (4)

or, since this implies R = 0,

Buy = O, (5)

we get

R ~uv ~ v " (6)

Contracting this equation with two vierbeins ha c' and hbO (the bein indices are denoted by latin letters) we get

Rabuv = Ra*~u v. (7)

The mixed curvature tensor can be expressed by means of(see e.g. [2])

Rabuv = Op .4 abu-- Ov.~ ab + [.4u, .4v]ab (8)

where the ,4's are the spin connections given by the standard formula

.~ ab =._ hbv ~)~ h a v _ r ; ° hao hbv

= - hbv Du hav, ,~ab = _ ,,~bau, (9)

where Du denotes the usual covariant derivative in Einstein's theory.

Now it is easily seen that

h p h d ha h ~ e a f 3 P ° = ~ e ahcd, (10)

from which

h a hb~ e c43pa = ~ e abcd hcP h r . (11)

Using eq. (1 1) in (6) we obtain

i [det ( -g) ] -1 /2 eabcd e~v6h (12) Rabp'v = 4 Rcdsh.

From this equation we obtain

a u ( X ~ ( -g ) R ably) + ~ ) [ . 4 "u' R"UVlab

• . eU,~XR - -"4- t eabed{au(eu~6XRcd~h)+[.~ u' ..Shlcd}"

(13)

From eq. (8) it follows that Rab~v can be expressed in terms of"potent ia ls" .~ab, and hence the fight hand side of eq. (13) vanishes, i.e.

V u ( ~ ) R-~ub) ___ 0 (14)

provided the potentials do not have Dirac string sin- gularities. In eq. (14), V u denotes the Yang-Mills

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Volume 71B, number 1 PHYSICS LETTERS 7 November 1977

covariant derivative (not to be confused with D u in eq. (9)), constructed in the usual way. From eqs. (8) and (14) it is seen that the assumption that the vacu- um Einstein equations are satisfied implies that the Yang-Mills equations are satisfied in curved space.

To conform with the usual notation we can intro- duce new potentials Aau b and field strengths b'~ b by the equations

ab 1 .~ ab Au : g ~, (15)

where g is the coupling constant (not to be confused with det (-g)) .

It is clear that a rather large number of solutions of the Yang-Mills equations (14) in curved space can be obtained just by borrowing the known solutions of the vacuum Einstein equations and constructing the potentials according to eq. (9). If one goes to flat space, these solutions clearly disappear.

If one consider the Einstein equations in the pres- ence of matter it follows from eq. (1) that the self- duality property (6) is lost. Therefore, the correspond- ing Yang-Mills equations in curved space contain a source. The source current can be expressed in terms of the energy-momentum tensor. There is therefore always a mapping from the Einstein equations to the curved space Yang-Mills equations. Since the curva- ture tensor has symmetry properties not shared by the Yang-Mills field strengths, we expect that the op- posite statement is not true.

! thank several members of the theoretical high energy group in Copenhagen for useful discussions.

References

[ 1 ] C. Lanczos, Ann. of Math. 39 (1938) 842. [2] B. De Wlt.t, Dynamical theory of groups and fields

(Blackie and Son, London and Glasgow, 1965).

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