Mass in general relativity
In general relativity, mass encompasses not only the rest masses of particles but all contributions from energy, momentum, pressure, and stress that source the curvature of spacetime via the Einstein field equations, where the stress-energy-momentum tensor T^{\mu\nu} plays the central role in coupling matter to geometry.[1] Unlike the singular, observer-independent mass of Newtonian gravity, mass in general relativity lacks a unique definition and instead requires context-specific formulations, such as global measures for isolated systems or quasi-local expressions for finite regions, reflecting the theory's diffeomorphism invariance and the absence of a fixed background.[1] Key definitions include the Arnowitt-Deser-Misner (ADM) mass, which quantifies the total energy of an isolated system in an asymptotically flat spacetime by evaluating a surface integral at spatial infinity using the Hamiltonian formulation of general relativity. This mass, introduced in the canonical analysis of the theory, is positive for physically reasonable matter distributions satisfying the dominant energy condition, as proven by the positive mass theorem, and includes binding energies that can make it smaller than the sum of individual constituent masses.[1] For stationary spacetimes admitting a timelike Killing vector, the Komar mass provides a conserved quantity derived from the covariant conservation laws, obtained as the surface integral (1/(4\pi)) \int_S *d\xi over a closed 2-surface, where \xi is the Killing 1-form and * denotes the Hodge dual, and it coincides with the ADM mass in asymptotically flat vacuum regions.[1][2] Further distinctions arise between inertial mass, which resists acceleration and corresponds to the ADM mass in certain limits, and active gravitational mass, which generates the field and aligns with definitions like the Møller mass; these are generally unequal inside matter distributions but equalize in vacuum or under energy conditions such as the weak or dominant ones.[3] For radiating systems, the Bondi mass accounts for energy loss via gravitational waves at null infinity, decreasing monotonically as radiation carries away mass-energy.[1] Quasi-local masses, such as the Hawking or Bartnik masses, attempt to localize energy within finite boundaries without reference to infinity, though they face challenges like gauge dependence and non-uniqueness.[1] These concepts underpin phenomena like black hole thermodynamics, where mass relates to entropy and temperature, and gravitational wave detection, where binary mergers reveal post-Newtonian corrections to mass motion.[1]Fundamental Concepts and Challenges
Core Concepts of Mass
In special relativity, the principle of mass-energy equivalence establishes that mass m and energy E are interchangeable, encapsulated by the relation E = mc^2, where c is the speed of light; this arises from the relativistic energy-momentum relation for a particle at rest. This equivalence implies that any form of energy contributes to the inertial mass of a system. In general relativity, this concept extends beyond point particles to fields and continuous distributions, where the stress-energy-momentum tensor T^{\mu\nu} serves as the fundamental source incorporating all forms of energy density, momentum flux, and stresses, generalizing the role of mass as an energy contributor.[4] The Einstein field equations, G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}, link the geometry of spacetime—described by the Einstein tensor G_{\mu\nu}, which encodes curvature—directly to the stress-energy tensor T^{\mu\nu} as the source term, with G denoting the gravitational constant.[4] Here, mass enters indirectly through the energy components of T^{\mu\nu}, such as the T^{00} term representing energy density, which generates gravitational effects equivalent to those of rest mass in the Newtonian limit; all forms of energy, including kinetic, potential, and field energies, thus curve spacetime.[5] In general relativity, conserved quantities like total energy emerge from Noether's theorem applied to spacetime symmetries, particularly Killing vectors representing asymptotic time translations or spatial translations in stationary or asymptotically flat spacetimes.[6] For matter fields, this yields a conserved four-momentum P^\mu = \int T^{\mu\nu} \, d\Sigma_\nu integrated over a spacelike hypersurface \Sigma, reflecting the invariance under these symmetries; however, the full gravitational energy requires accounting for the pseudotensor contributions from the metric.[6] The Arnowitt-Deser-Misner (ADM) formalism provides a global notion of energy by defining the total four-momentum via a surface integral at spatial infinity in asymptotically flat spacetimes, capturing the system's overall energy without local densities.[6] In general relativity, the term "mass" typically denotes the total energy E of an isolated system as measured at infinity, encompassing not only the rest masses and internal energies of constituents but also the negative gravitational binding energy, which reduces the overall value compared to the sum of unbound parts.[7] This binding contribution arises from the self-gravitational interaction, making the total mass a measure of the system's irreducible energy content.[7]Obstacles to Definition
In general relativity, the equivalence principle asserts that locally, the effects of gravity are indistinguishable from those of acceleration, implying no unique local distinction between gravitational and inertial mass. This principle, central to the theory, necessitates non-local definitions for mass, as any attempt to measure mass purely through local observations would conflate gravitational influences with inertial ones.[8] Attempts to define a local energy density for gravity, such as through pseudo-tensors, encounter significant obstacles due to their coordinate dependence and lack of covariance. The Landau-Lifshitz pseudo-tensor, for instance, provides an expression for gravitational energy-momentum in terms of the metric perturbation g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, but it transforms non-tensorially under general coordinate changes, rendering it unsuitable for a coordinate-independent local mass definition.[6][9] Positive energy theorems highlight further constraints on mass definitions, requiring specific conditions like non-negative matter energy density to ensure the total mass is non-negative. The Schoen-Yau theorem, for example, proves that for asymptotically flat spacetimes satisfying the dominant energy condition, the ADM mass is positive unless the spacetime is flat Minkowski space. These results underscore the need for global constraints to avoid negative or ill-defined local masses. In curved spacetime, energy is not conserved locally owing to the diffeomorphism invariance of general relativity, which precludes a unique timelike Killing vector for defining conserved quantities everywhere; conservation holds only globally in stationary spacetimes or asymptotically. This non-local nature of energy conservation complicates any local mass notion, as gravitational energy cannot be isolated without reference to the overall spacetime structure.[6][10] A illustrative example is the Schwarzschild metric, which describes the exterior geometry of a spherically symmetric, non-rotating mass:ds^2 = -\left(1 - \frac{2M}{r}\right) dt^2 + \left(1 - \frac{2M}{r}\right)^{-1} dr^2 + r^2 d\Omega^2,
where M is a parameter with dimensions of mass, interpretable as the total mass but not directly measurable through local experiments, as it emerges only from the global asymptotic behavior of the spacetime.[11]