Black holes and wormholes in Deser-Woodard gravity

TL;DR

This work develops fully analytic static vacuum solutions in the revised Deser--Woodard nonlocal gravity by employing a special tetrad frame that reduces the field equations to a tractable form. Using Schwarzschild and Reissner--Nordström seeds, the authors construct black-hole, wormhole, and naked-singularity spacetimes as deformations characterized by an integration constant $C$, compute the nonlocal auxiliary fields, and analyze horizon structure and energy-condition implications. They show that regular black holes arise only from extremal RN seeds, while traversable wormholes emerge for appropriate $C$ and $C<C_0$, with auxiliary fields remaining finite for wormhole spacetimes; horizon-related divergences can occur for some regular-bh solutions. Shadow calculations reveal that some Deser--Woodard geometries cast shadows larger than Schwarzschild, providing potential observational discriminants for nonlocal gravity at strong gravity regimes.

Abstract

The Deser-Woodard gravity is a modified theory of gravity in which nonlocality plays a central role. In this context, nonlocality is a consequence of the inverse of the d'Alembertian operator $\square^{-1}$ in the effective action. Here, exact black hole and wormhole solutions are built in the revised Deser-Woodard gravity following a recent approach, where a special tetrad frame simplifies the complicated field equations of the theory. Using the Schwarzschild metric and the Reissner-Nordström metric as initial seed solutions, the developed algorithm generates new traversable wormholes, singular black holes and a regular black hole as solutions of the vacuum field equations of the modified theory. Also, the auxiliary fields, which are responsible for the nonlocality, are computed. However, even for a regular black hole solution, in which spacetime does not contain a curvature singularity, the corresponding auxiliary fields diverge at the event horizon. Regarding observational results, the shadow angular radius is computed for the new solutions. In particular, the deviation of the Schwarzschild black hole in the Deser-Woodard gravity casts a larger shadow than the corresponding black hole in general relativity.

Figures (7)

• Figure 1: The metric functions given by Eqs.(\ref{['A_Schw']})-(\ref{['B_Schw']}) for the Schwarzschild deviation with different values of the integration constant $C$. As one adopts $M=1$, the vertical dashed line indicates the curvature singularity at $r_0=3$, and the event horizon of the Schwarzschild black hole is located at $r=2$.
• Figure 2: The metric functions given by Eqs.(\ref{['A_RN']})-(\ref{['B_RN']}) for the Reissner--Nordström deviation with different values of the integration constant $C$. As one adopts $M=1$ and $Q=0.1$ (nonextremal), the vertical dashed line indicates the curvature singularity $r_0=2.99$ (nonextremal case). The event horizon of the Reissner--Nordström black hole is located at $r_+=1.99$ (nonextremal) and at $r_+=1$ (extremal). The smaller root of $B(r)$ is $r_- = 0.83$ for the extremal case (see the enlarged image).
• Figure 3: Auxiliary fields for the Morris--Thorne solution (\ref{['MT']}) with $\mathcal{C}=-1$. The vertical dashed line indicates the wormhole throat. As we can see, all fields are finite at the throat.
• Figure 4: The tensor component $\mathcal{K}_{rr}$, which depends on the auxiliary fields, calculated directly from the fields equations, approximately and numerically. As one adopts $M=1$, the vertical dashed line indicates the event horizon. As we can see, $\mathcal{K}_{rr}$ diverges at this point. For the approximate result, one adopts $X_0 =Y_0 = V_0 =2$. And for the numerical calculations, one assumes the asymptotic behavior of the auxiliary fields.
• Figure 5: Auxiliary fields for the regular black hole solution (\ref{['RBH']}) numerically calculated. The vertical dashed line indicates the event horizon. As we can see, all fields diverges at the event horizon. In this graphic, one adopts $M=1$.