Birkhoff’s theorem in Lovelock gravity for general base manifolds

Birkhoff's theorem in higher dimensions is valid for a wider class of metrics which include spacetimes with Böhm metrics existing on base manifolds topologically spheres or products thereof.

“Hence, Birkhoff's theorem in higher dimensions is valid for a wider class of metrics which in particular include spacetimes with the Böhm metrics existing on base manifolds which are topologically spheres or products thereoffoot_0 [1][2][3].”

• cites [1] Böhm, 1998, Invent. Math. sourceNot checked — no full text of the source
• cites [2] Gibbons, 2003, Prog. Theor. Phys. Suppl. sourcePartly supported

  Section 3 (page unavailable) describes higher-dimensional, non-asymptotically flat static black-hole solutions obtained by using Böhm metrics in place of round sphere metrics. It does not establish the claimed extension of Birkhoff’s theorem or state that the Böhm metrics occur on products of spheres. (LLM, conf=0.84)

  Experimental — accuracy not yet measured
• cites [3] Gibbons, 2002, Phys. Rev. Lett. sourcePartly supported

  The concluding discussion (p. 4, around Eq. 40) says that replacing a round-sphere metric with Böhm Einstein metrics yields non-asymptotically flat static black-hole solutions. It does not establish Birkhoff’s theorem for that wider class; it expressly leaves the uniqueness of black holes constructed from the same Böhm metric open. (LLM, conf=0.86)

  Experimental — accuracy not yet measured

These choices may even allow non-static solutions.

“These choices may even allow non-static solutions [11,14].”

• cites [11] Dotti, 2010, Phys. Rev. D sourceSupported
• cites [14] Oliva, 2013, J. Math. Phys. sourceSupported

Birkhoff's theorem also holds in Lovelock theory.

“Birkhoff's theorem also holds in Lovelock theory of gravity [4,5] which is a natural higher curvature generalization of Einstein's general relativity and retains the property that the field equations are of second order in the metric [6].”

• cites [4] Zegers, 2005, J. Math. Phys. sourcePartly supported

  The theorem on p. 1 establishes a Birkhoff-type result for generic Lovelock field equations with spherical, planar, or hyperbolic symmetry. The qualification matters: the paper says fine-tuned Lovelock couplings can evade the theorem, so the unqualified claim is broader than its result. (LLM, conf=0.88)

  Experimental — accuracy not yet measured
• cites [5] Deser, 2005, Class. Quantum Grav. sourceSupported

Any solution of Lovelock gravity of the form of a warped product of a two-dimensional transverse space and an arbitrary base manifold must be static.

“In particular, it is shown that any solution of the form of a warped product of a two-dimensional transverse space and an arbitrary base manifold must be static.”

• Also stated in: § 2. Staticity and reduced field equations · plain

  “This implies that the general solution of the form (1) of the field equations (3) must be static and reduces to”

The metric in the transverse space is determined by a single function of a spacelike coordinate which satisfies an algebraic equation involving the constants characterizing the base manifold along with the coupling constants.

“and the metric in the transverse space is determined by a single function of a spacelike coordinate which satisfies an algebraic equation involving the constants characterizing the base manifold along with the coupling constants.”

• Also stated in: § 4. Conclusion · boosted

  “Using a simple method we have shown that for generic coupling constants, the transverse part of the metric is given in terms of a single function of the spacelike coordinate r which is determined by an algebraic equation.”

All Euler invariants of the base manifold must be constants.

“Moreover, since 1, 0 ˆ( )  = they all must also be constants.”

The constants alpha_n generally depend on the geometry of the base manifold.

“However, the constants n a ʼs generally depend on the geometry of the base manifold as we shall show now.”

For the maximum multiplicity choice of coupling constants, the metric function is given by g(r) = r^2 U = ... const.

“Hence the metric function in this case is given by g r r U r r const. . D k k 2 2 2 1 ( ) ( ) ¯b g = - = + - --”

The algebraic equation (8) can be solved explicitly for g(r) only up to k=4 for generic values of the coupling constants.

“The algebraic equation (8) can be solved explicitly for g(r) only up to k 4 ¯= for generic values of the coupling constants.”

All Euler invariants of the base manifold must be multiples of each other.

“In other words, all the Euler invariants of the base manifold must be multiples of each other.”

In the maximum multiplicity case, the algebraic equation has a unique solution if and only if all coupling constants a_n and base constants alpha_n are related to parameters beta and gamma through specific relations.

“Furthermore, observing the highest order term in U, we infer that the equation has a unique solution if and only if the left hand side can be expressed as a U r . . n m k k m n ¯   a b g = - ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Putting m k = we conclude that all the coupling constants a n and those characterizing the base manifold n a must be related to the parameters β and γ through the relations a a k k and , n n k k”

The field equations restrict the base manifold such that all non-trivial intrinsic Lovelock tensors of the base manifold are constants.

“Furthermore, the field equations are shown to dictate that all the Lovelock tensors and the Euler invariants up to order k ¯of the base manifold are constants, where k ¯is the highest order of the curvature terms in the action.”

There can be at most k distinct solutions to equation (8) depending on the coupling constants a_n and the constants characterizing the base manifold alpha_n.

“There can be at most k ¯distinct solutions to the equation (8) depending on the coupling constants a n and the constants characterizing the base manifold . n a”

For a non-trivial solution of the functional equation to exist, the sets of functions {f_k(x)} and {g_k(y)} must be linearly dependent.

“The basic idea is that for a non-trivial solution to exist, the sets of functions f x k { ( )} and g y k { ( )} must be linearly dependent sets.”

• cites [19] Aczél, 2006Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [20] Sutô, 1914, Tôhoku Math. J.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Generically the constants alpha_n characterizing the base manifold, along with the coupling constants c_n, determine the asymptotic structure of the corresponding solution.

“Moreover, generically the constants n a characterizing the base manifold, along with the coupling constants c n , determine the asymptotic structure of the corresponding solution.”

The Schwarzschild-Tangherlini metric is the unique asymptotically flat spacetime in the whole family.

“Note however that the Schwarzschild-Tangherlini metric is the unique asymptotically flat spacetime in the whole family.”

Equation (8) is the generalization of the polynomial equation first given by Wheeler, whose solution determines the metric function for spherically symmetric solutions of Lovelock gravity.

“This is the generalization of the polynomial equation, first given by Wheeler [18], whose solution determines the metric function for spherically symmetric solutions of Lovelock gravity.”

• cites [18] Wheeler, 1986, Nucl. Phys. B sourceNot checked — no full text of the source

The constants characterizing the base manifold can be chosen arbitrarily.

“all the non-trivial intrinsic Lovelock tensors of the base manifold are constants, which can be chosen arbitrarily”

• Also stated in: § 4. Conclusion · plain

  “These constants (if not identically zero) can be chosen arbitrarily but once fixed they do appear in the metric function in the transverse space since they must satisfy the algebraic equation (8).”