ADM Energy–Momentum and Its Hypotheses

ADM energy–momentum is a charge of an asymptotically Euclidean end, not a coordinate formula that may be evaluated in an arbitrary chart. Its surface integrals are finite and geometric only after decay, differentiability, integrability, and allowed-coordinate hypotheses have been fixed. The dominant energy condition is not needed to write the charge, but it is the hypothesis that turns the charge into the future-causal quantity of the positive mass theorem. This page separates those two roles and checks the normalization on a Schwarzschild slice.

Required background. Analysis for Mathematical Relativity supplies weighted spaces, Hypersurfaces and Foliations supplies initial-data geometry, Curvature supplies scalar curvature, and Einstein Equations supplies the constraints.

Helpful background. Causality and Global Structure clarifies the spacetime meaning of asymptotic ends, while Partial Differential Equations and Green Functions supports the elliptic estimates used in invariance proofs.

Asymptotically Euclidean initial data

Let $(\Sigma,h,K)$ be a complete three-dimensional initial data set with an end diffeomorphic to $\mathbb R^3$ minus a ball. In an admissible asymptotic chart $x^i$, write $r=|x|$ and require, for some $q>1/2$ and sufficient weighted regularity,

$$h_{ij}=\delta_{ij}+O_2(r^{-q}), \qquad K_{ij}=O_1(r^{-1-q}).$$

Here $O_k$ controls derivatives through order $k$ with the corresponding extra inverse powers of $r$. One also requires integrability of the constraint sources, commonly $R(h)\in L^1$ and suitable integrability of $D^j(K_{ij}-Kh_{ij})$. These conditions suppress chart-dependent leading terms that a bare pointwise falloff would not control.

The site’s convention is

$$K_{ij}=-h_i{}^\mu h_j{}^\nu\nabla_\mu n_\nu,$$

for future unit normal $n$. With this sign the constraint densities are

$$16\pi G\rho=R(h)+K^2-K_{ij}K^{ij}, \qquad 8\pi G j_i=D^j(K_{ij}-Kh_{ij}).$$

Sources using the opposite sign for $K$ reverse the displayed momentum convention and must be translated as a whole.

The threshold $q>1/2$ is tied to three spatial dimensions. Quadratic errors in the scalar-curvature expansion then decay integrably once the stated weighted regularity and $R(h)\in L^1$ condition are imposed. The linear part of $R(h)$ is a Euclidean divergence whose volume integral becomes the ADM surface term. Under an admissible change of asymptotic coordinates, the change in that linear divergence is itself a boundary divergence plus terms controlled by the integrable curvature. This is the core of the invariance proof.

Pointwise notation alone hides these facts. Saying only $h-\delta=o(1)$, for example, permits derivatives that oscillate too strongly for the surface integral to converge. Conversely, requiring an unnecessarily rigid Schwarzschild expansion would exclude valid asymptotically Euclidean data. Weighted Sobolev or Hölder formulations balance these demands by controlling both decay and differentiability; the chosen formulation must be stated before one compares masses computed in different charts.

Energy and momentum surface integrals

Let $S_r$ be a large coordinate sphere with Euclidean outward normal $\nu^i$ and area element $dS$. The ADM energy and spatial momentum are

$$E_{\mathrm{ADM}} =\frac{1}{16\pi G}\lim_{r\to\infty} \int_{S_r}(\partial_jh_{ij}-\partial_i h_{jj})\nu^i\,dS,$$ $$P_i =\frac{1}{8\pi G}\lim_{r\to\infty} \int_{S_r}(K_{ij}-Kh_{ij})\nu^j\,dS.$$

The limits are defined end by end. Under asymptotic Euclidean changes of chart compatible with the weighted hypotheses, $E_{\mathrm{ADM}}$ is invariant and $P_i$ transforms as the spatial part of asymptotic energy–momentum. Bartnik’s weighted analysis identifies the coordinate-invariant mass under sharp integrability assumptions; see Bartnik 1986, Theorem 4.2 and §§3–4, pp. 675–686.

Regge–Teitelboim parity conditions are especially important for center of mass and angular momentum, whose integrals carry an additional factor of $x$. They are not a decorative strengthening of the energy definition. A page quoting all Poincaré charges must state decay and parity separately rather than imply that existence of ADM energy automatically gives every other charge.

Constraint and positivity hypotheses

For matter data, the dominant energy condition on the slice is

$$\rho\ge\sqrt{h^{ij}j_i j_j}.$$

It implies that the stress–energy flux measured by the normal is causal. In the positive mass theorem it yields $E_{\mathrm{ADM}}\ge\sqrt{\delta^{ij}P_iP_j}$. Completeness, asymptotic flatness, dimension, regularity, and boundary hypotheses remain independent; the energy condition alone cannot repair a missing end or singular interior. Schoen and Yau’s 1979 result treats the time-symmetric case $K=0$; the full spacetime inequality under these hypotheses is Eichmair, Huang, Lee, and Schoen 2016, Theorem 1 and §2, pp. 83–90. The independent spinorial formulation is Witten 1981, §§2–3, pp. 384–391.

Schwarzschild normalization check

On a time-symmetric Schwarzschild end in isotropic radius,

$$h_{ij}=\left(1+\frac{GM}{2r}\right)^4\delta_{ij}, \qquad K_{ij}=0.$$

Thus $h_{ij}=(1+2GM/r+O(r^{-2}))\delta_{ij}$. The energy integrand is $4GM/r^2+O(r^{-3})$ after contraction with $\nu^i$, so integration over $S_r$ and division by $16\pi G$ gives $E_{\mathrm{ADM}}=M$. The momentum vanishes because $K=0$. This check fixes both the factor of $16\pi G$ and the interpretation of $M$ as physical mass rather than geometrized length.

Limitations and bad coordinates

The surface expression is not invariant under arbitrary transformations that approach the identity too slowly. Flat Euclidean data written in a badly behaved asymptotic chart can produce nonconvergent or spurious-looking boundary terms; this is a counterexample to defining mass from the formula without an admissible chart class. Multiple ends can also carry different ADM vectors.

Here is an explicit borderline example. Start with Euclidean space in ordinary radius $r$ and introduce a new radius $R$ by $r=R+b\sqrt R$ for large $R$, where $b\ne0$ has dimensions of length$^{1/2}$. In Cartesian coordinates $X^i=Rn^i$ the same flat metric is

$$h_{ij}=a(R)n_i n_j+c(R)(\delta_{ij}-n_i n_j), \qquad a=\left(1+\frac{b}{2\sqrt R}\right)^2, \quad c=\left(1+\frac{b}{\sqrt R}\right)^2.$$

Direct differentiation gives

$$(\partial_jh_{ij}-\partial_i h_{jj})n^i =2\left(\frac{a-c}{R}-c'\right) =\frac{b^2}{2R^2}.$$

The surface formula would therefore return $b^2/(8G)$ for a flat metric! There is no contradiction: this chart has only $q=1/2$, precisely the excluded decay threshold. The original Euclidean chart gives zero. Scalar curvature integrability alone cannot rescue an inadmissible asymptotic chart.

For an end with nonzero $P_i$, the invariant rest mass is $m_{\mathrm{ADM}}=\sqrt{E_{\mathrm{ADM}}^2-\delta^{ij}P_iP_j}$ once the positive mass theorem has shown that the ADM four-vector is future causal. Before that theorem, the square root need not be known real. Energy, momentum, and rest mass therefore should not be used as interchangeable names. A change of asymptotic inertial frame changes the components $(E,P_i)$ while leaving the Lorentzian norm fixed.

A further limitation is that ADM energy is global at spatial infinity. It is not a quasi-local mass of a finite surface, and it need not be meaningful for asymptotically anti-de Sitter or cosmological data without a different charge definition and background subtraction. The proof of coordinate invariance uses the constraint and weighted hypotheses; it does not follow from formal index notation.

Check the Schwarzschild end

Use $h_{ij}=f\delta_{ij}$ with $f=1+2GM/r+O(r^{-2})$ to evaluate the leading ADM energy integrand. Why does the result not depend on the $O(r^{-2})$ terms?

Solution

One has $\partial_jh_{ij}=\partial_i f$ and $\partial_i h_{jj}=3\partial_i f$, so their difference is $-2\partial_i f=4GMx_i/r^3+O(r^{-3})$. Contracting with $\nu^i=x^i/r$ gives $4GM/r^2+O(r^{-3})$. The leading term integrates to $16\pi GM$; the remainder integrates to $O(r^{-1})$ and vanishes. Multiplication by $1/(16\pi G)$ yields $M$.

References

• Bartnik, Robert. “The Mass of an Asymptotically Flat Manifold.” Communications on Pure and Applied Mathematics 39 (1986): 661–693. DOI.
• Eichmair, Michael, Lan-Hsuan Huang, Dan A. Lee, and Richard Schoen. “The Spacetime Positive Mass Theorem in Dimensions Less Than Eight.” Journal of the European Mathematical Society 18 (2016): 83–121. DOI.
• Witten, Edward. “A New Proof of the Positive Energy Theorem.” Communications in Mathematical Physics 80 (1981): 381–402. DOI.