Smarr Relations from Scaling

A Smarr relation is an algebraic identity among the charges and horizon data of one stationary black hole. In four-dimensional asymptotically flat Einstein–Maxwell theory it follows from the first law plus homogeneity under a uniform rescaling of all lengths. The derivation makes its limitations transparent: changing dimension changes the weights, while a cosmological constant, higher-curvature coupling, or other dimensionful parameter adds a new term unless it is held fixed outside the scaling family.

Required background. Variational and Symplectic Geometry supplies the Hamiltonian first law. Helpful background. Exact Spacetimes supplies Kerr–Newman checks; Causality and Global Structure fixes asymptotic ends; Einstein Equations fixes the theory and couplings; Gravitational Actions and Boundary Terms supplies the variational boundaries; Black-Hole Geometry and Mechanics supplies normalized horizon data.

Four-dimensional scaling weights

Consider a smooth family of stationary, asymptotically flat, four-dimensional Einstein–Maxwell black holes. Keep $G$ and the Maxwell normalization fixed. Under a uniform rescaling of lengths by $\lambda$,

$$A_H\mapsto\lambda^2A_H, \qquad J\mapsto\lambda^2J, \qquad Q\mapsto\lambda Q, \qquad M\mapsto\lambda M.$$

The charge weight follows from the metric combination $GQ^2/(4\pi r^2)$; the angular-momentum weight follows from $GJ$ having dimension length squared. Therefore the mass function obeys

$$M(\lambda^2A_H,\lambda^2J,\lambda Q)= \lambda M(A_H,J,Q).$$

Differentiate with respect to $\lambda$ at $\lambda=1$. Euler’s theorem gives

$$M=2A_H\frac{\partial M}{\partial A_H} +2J\frac{\partial M}{\partial J} +Q\frac{\partial M}{\partial Q}.$$

For variations tangent to the stationary solution family, the first law identifies

$$\frac{\partial M}{\partial A_H}=\frac{\kappa}{8\pi G}, \qquad \frac{\partial M}{\partial J}=\Omega_H, \qquad \frac{\partial M}{\partial Q}=\Phi_H.$$

Consequently,

$$M=\frac{\kappa A_H}{4\pi G}+2\Omega_HJ+\Phi_HQ.$$

Bardeen, Carter, and Hawking derive this stationary mass formula and its variation in Bardeen, Carter, and Hawking 1973, Open PDF, pp. 163–166. The factor two multiplying $\Omega_HJ$ is a scaling weight, consistent with the factor in the Komar integral.

Independent Kerr–Newman check

Introduce the irreducible mass by

$$M_{\mathrm{irr}}^2=\frac{A_H}{16\pi G^2}.$$

With the rationalized electric charge used here, the Kerr–Newman horizon data obey the Christodoulou–Ruffini relation Christodoulou and Ruffini 1971, pp. 3552–3555:

$$M^2= \left(M_{\mathrm{irr}} +\frac{Q^2}{16\pi G M_{\mathrm{irr}}}\right)^2 +\frac{J^2}{4G^2M_{\mathrm{irr}}^2}.$$

Every term on the right has dimensions of mass squared. Under the preceding length rescaling, $M_{\mathrm{irr}}$, $Q$, and $\sqrt J$ all have weight one, so the formula is homogeneous. Differentiating it identifies the coefficients of $\delta A_H$, $\delta J$, and $\delta Q$ with the Kerr–Newman $\kappa/(8\pi G)$, $\Omega_H$, and $\Phi_H$. Euler’s theorem then returns the displayed Smarr formula independently of the Komar boundary identity.

Three limits discriminate common errors. Setting $J=Q=0$ yields $M=\kappa A_H/(4\pi G)$. Setting $Q=0$ retains the factor $2\Omega_HJ$. Taking the extremal limit sets $\kappa=0$ but leaves $M=2\Omega_HJ+\Phi_HQ$; it does not set all variations or all work terms to zero.

Dimension changes the coefficients

In $D$ spacetime dimensions with fixed Newton constant, mass and electric charge have length weight $D-3$, while horizon area and angular momentum have weight $D-2$. Applying Euler homogeneity to the corresponding first law gives

$$(D-3)M =\frac{(D-2)\kappa A_H}{8\pi G_D} +(D-2)\sum_i\Omega_iJ_i +(D-3)\Phi_HQ$$

for an asymptotically flat Einstein–Maxwell family with the stated normalization and no additional scales. In $D=4$ this reduces to the previous formula. Copying the four-dimensional coefficients into a Myers–Perry or Tangherlini calculation would fail even before any detailed metric calculation.

The precise charge dimension can depend on how the Maxwell coupling is written. The invariant procedure is to read every weight from the declared action and physical dimensions, not memorize the displayed formula while changing conventions.

Cosmological pressure and new scales

Let the cosmological constant vary within an extended phase space and define

$$P=-\frac{\Lambda}{8\pi G}, \qquad \delta M=\frac{\kappa}{8\pi G}\delta A_H +\Omega_H\delta J+\Phi_H\delta Q+V\delta P.$$

In four dimensions, $P$ has length weight $-2$. Euler’s theorem therefore adds $-2PV$:

$$M=\frac{\kappa A_H}{4\pi G} +2\Omega_HJ+\Phi_HQ-2PV.$$

Kubizňák, Mann, and Teo formulate this extended scaling and its thermodynamic volume in Kubizňák, Mann, and Teo 2017, §2.2, equations (2.14)–(2.21), pp. 7–9. The formula does not assert that $\Lambda$ must vary physically. It describes a chosen solution space whose tangent vectors include $\delta\Lambda$. If the phase space fixes $\Lambda$, the first-law variation has no $V\delta P$ term even though the algebraic Smarr identity can still contain the scale through a geometric volume term.

A higher-curvature coupling $\alpha$ of length weight two, a string tension, or a conical deficit produces an analogous contribution when varied. If a dimensionful coupling is present but omitted from the homogeneity relation, $M$ is not homogeneous in the listed extensive variables and the short Euler derivation is invalid. One may instead derive an identity from Noether or Komar methods, provided all bulk and boundary terms are retained.

Smarr is not an integrated first law by default

The first law compares neighboring solutions and is a one-form on solution space. The Smarr relation holds at one solution because of scaling. Integrating the first law along an arbitrary path does not generally produce Smarr: the intensive coefficients vary, and integrability must first be established. Conversely, differentiating an algebraic Smarr formula can produce extra terms unless the quantities satisfy the correct equations of state.

This distinction becomes important for several horizons or variable couplings. Each horizon may have its own generator and surface gravity, while global constraints can prevent their areas from being independently varied. The permitted tangent space must be declared before Euler’s theorem is applied.

Derive the Schwarzschild–AdS term

For a four-dimensional static AdS black hole with pressure $P$, suppose the extended first law contains only $A_H$ and $P$. Use their scaling weights to derive the Smarr relation. Then take $P\to0$.

Solution

Mass has weight one, area weight two, and pressure weight minus two. Euler’s theorem gives

$$M=2A_H\frac{\partial M}{\partial A_H} -2P\frac{\partial M}{\partial P} =\frac{\kappa A_H}{4\pi G}-2PV.$$

When $P\to0$ along a family whose horizon data approach Schwarzschild, the second term vanishes and $M=\kappa A_H/(4\pi G)$. Omitting the negative weight of $P$ would give the wrong sign and would not reproduce the known geometric identity.

Covariant Phase Space and Horizon Hamiltonians next derives the variational identity and its integrability conditions rather than assuming a mass function.

References

• Bardeen, James M., Brandon Carter, and Stephen W. Hawking. “The Four Laws of Black Hole Mechanics.” Communications in Mathematical Physics 31 (1973): 161–170. DOI.
• Christodoulou, Demetrios, and Remo Ruffini. “Reversible Transformations of a Charged Black Hole.” Physical Review D 4 (1971): 3552–3555. DOI.
• Kubizňák, David, Robert B. Mann, and Mae Teo. “Black Hole Chemistry: Thermodynamics with Lambda.” Classical and Quantum Gravity 34 (2017): 063001. DOI.