We give a derivation of general relativity (GR) and the gauge principle that is novel in presupposing neither spacetime nor the relativity principle. We consider a class of actions defined on superspace (the space of Riemannian 3-geometries on a given bare manifold). It has two key properties. The first is symmetry under 3-diffeomorphisms. This is the only postulated symmetry, and it leads to a constraint linear in the canonical momenta. The second property is that the Lagrangian is constructed from a 'local' square root of an expression quadratic in the velocities. The square root is 'local' because it is taken before integration over 3-space. It gives rise to quadratic constraints that do not correspond to any symmetry and are not, in general, propagated by the Euler-Lagrange equations. Therefore these actions are internally inconsistent. However, one action of this form is well behaved: the Baierlein-Sharp-Wheeler (BSW [1]) reparametrisation-invariant action for GR. From this viewpoint, spacetime symmetry is emergent. It appears as a 'hidden' symmetry in the (underdetermined) solutions of the Euler-Lagrange equations, without being manifestly coded into the action itself. In addition, propagation of the linear diffeomorphism constraint together with the quadratic square-root constraint acts as a striking selection mechanism beyond pure gravity. If a scalar field is included in the configuration space, it must have the same characteristic speed as gravity. Thus Einstein causality emerges. Finally, self-consistency requires that any 3-vector field must satisfy Einstein causality, the equivalence principle and, in addition, the Gauss constraint. Therefore we recover the standard (massless) Maxwell equations.

When constructing general relativity (GR), Einstein required 4D general covariance. In contrast, we derive GR (in the compact, without boundary case) as a theory of evolving 3-dimensional conformal Riemannian geometries obtained by imposing two general principles: 1) time is derived from change; 2) motion and size are relative. We write down an explicit action based on them. We obtain not only GR in the CMC gauge, in its Hamiltonian 3 + 1 reformulation but also all the equations used in York's conformal technique for solving the initial-value problem. This shows that the independent gravitational degrees of freedom obtained by York do not arise from a gauge fixing but from hitherto unrecognized fundamental symmetry principles. They can therefore be identified as the long-sought Hamiltonian physical gravitational degrees of freedom.Since Einstein created GR 4D spacetime covariance has been taken as its axiomatic basis. However, much work has been done in a dynamical approach that uses the 3+1 split into space and time of Arnowitt, Deser, and Misner (ADM) [1]. This work has been stimulated by the needs of astrophysics (especially gravitational-wave research) and by the desire to find a canonical version of GR suitable for quantization.The ADM formalism describes constrained Hamiltonian evolution of 3D spacelike hypersurfaces embedded in 4D spacetime. The intrinsic geometry of the hypersurfaces is represented by a Riemannian 3-metric g ij , which is the ADM canonical coordinate. The corresponding canonical momentum π ij is related to the extrinsic curvature κ ij of the embedding of the hypersurfaces in spacetime by π ij = − √ g(κ ij − g ij κ). The ADM dynamics, which respects full relativity of simultaneity by allowing free choice of the 3+1 split, is driven by two constraints. The linear momentum constraint π ij ;j = 0 reflects the gauge symmetry under 3D diffeomorphisms and is well understood. When it has been quotiented out, the 3 × 3 symmetric matrix g ij has three degrees of freedom. The quadratic Hamiltonian constraint gH = −π ij π ij + π 2 /2 + gR = 0 reflects the relativity of simultaneity -the time coordinate can be freely chosen at each space point. It shows that g ij has only two physical degrees of freedom. The problem is to find them. The solution, if it exists, will break 4D covariance.An important clue was obtained by York [2], who perfected Lichnerowicz's conformal technique [3] for finding initial data that satisfy the initial-value constraints of GR. In the Hamiltonian formalism, these are the ADM Hamiltonian and momentum constraints. Finding such data is far from trivial. York's is the only known effective method. He divides the 6 degrees of freedom in the 3-metric into three groups; 3 are mere coordinate freedoms, 1 is a scale part (a conformal factor), and the two remaining parts represent the conformal geometry, the 'shape of space'. York's method also introduces a distinguished foliation of spacetime -and with it a definition of simultaneity -by hypersurfaces of constant mean (extrinsic) curva...

We analyze the observational and theoretical constraints on "Einstein-aether theory", a generally covariant theory of gravity coupled to a dynamical, unit, timelike vector field that breaks local Lorentz symmetry. The results of a computation of the remaining post-Newtonian parameters are reported. These are combined with other results to determine the joint post-Newtonian, vacuumCerenkov, nucleosynthesis, stability, and positive-energy constraints. All of these constraints are satisfied by parameters in a large two-dimensional region in the four-dimensional parameter space defining the theory. * bzf@umd.edu † jacobson@umd.edu 1 Alternative theories of gravity that deviate from general relativity have been ruled out or severely constrained systematically as observations have improved [1,2]. At this stage the most studied surviving alternative is "scalar-tensor theory", of which Jordan-Brans-Dicke is a well-known example. This sort of theory is a simple extension of general relativity containing a fundamental scalar field. The existence of such scalar fields has been suggested by moduli fields that arise in higher dimensional approaches to gravity and quantum gravity, such as string theory.Vector-tensor theories consisting of general relativity coupled to a dynamical vector field are much more complicated than scalar-tensor theories, due to the metric derivatives that appear in the covariant derivative of the vector field. They also appear to suffer from a glaring problem: because of the indefinite signature of the spacetime metric, some of the degrees of freedom are always associated with negative energies. This problem need not occur, however, if the vector field is constrained to have a fixed timelike magnitude. Such a vector field specifies a particular rest frame at each point of spacetime; hence, it "spontaneously" breaks the local Lorentz symmetry in a dynamical fashion. These theories-of which there is a four-parameter family-preserve the full diffeomorphism symmetry group of general relativity, and gravity is still described by the curvature of the spacetime metric. The existence of such a vector field is not as well-motivated as a scalar field, however a number of approaches to quantum gravity have very tentatively suggested that Lorentz symmetry might be broken. If general covariance is to be preserved, then any Lorentz violating vector (or tensor) field must be dynamical.In this paper we discuss the complete collection of currently available observational and theoretical constraints on unit-vector-tensor theories, combining our new result for the remaining post-Newtonian parameters with previously established constraints. Surprisingly, all of these constraints are compatible with ranges of order unity for two coefficients in the Lagrangian. We are aware of no other theory that comes this close to so many predictions of general relativity and yet is fundamentally different.We call the timelike unit vector the aether, and the coupled theory Einstein-aether theory, or ae-theory for short. A revie...

We present a scale-invariant theory, conformal gravity, which closely resembles the geometrodynamical formulation of general relativity (GR). While previous attempts to create scaleinvariant theories of gravity have been based on Weyl's idea of a compensating field, our direct approach dispenses with this and is built by extension of the method of best matching w.r.t scaling developed in the parallel particle dynamics paper by one of the authors. In spatially-compact GR, there is an infinity of degrees of freedom that describe the shape of 3-space which interact with a single volume degree of freedom. In conformal gravity, the shape degrees of freedom remain, but the volume is no longer a dynamical variable. Further theories and formulations related to GR and conformal gravity are presented.Conformal gravity is successfully coupled to scalars and the gauge fields of nature. It should describe the solar system observations as well as GR does, but its cosmology and quantization will be completely different.

Abstract"Einstein-aether" theory is a generally covariant theory of gravity containing a dynamical preferred frame. This article continues an examination of effects on the motion of binary pulsar systems in this theory, by incorporating effects due to strong fields in the vicinity of neutron star pulsars. These effects are included through an effective approach, by treating the compact bodies as point particles with nonstandard, velocity dependent interactions parametrized by dimensionless "sensitivities". Effective post-Newtonian equations of motion for the bodies and the radiation damping rate are determined. More work is needed to calculate values of the sensitivities for a given fluid source; therefore, precise constraints on the theory's coupling constants cannot yet be stated. It is shown, however, that strong field effects will be negligible given current observational uncertainties if the dimensionless couplings are less than roughly 0.01 and two conditions that match the PPN parameters to those of pure general relativity are imposed. In this case, weak field results suffice. There then exists a one-parameter family of Einstein-aether theories with "small-enough" couplings that passes all current observational tests. No conclusion can be reached for larger couplings until the sensitivities for a given source can be calculated.

This work concerns the loss of energy of a material system due to gravitational radiation in Einstein-aether theory-an alternative theory of gravity in which the metric couples to a dynamical, timelike, unit-norm vector field. Derived to lowest post-Newtonian order are waveforms for the metric and vector fields far from a nearly Newtonian system and the rate of energy radiated by the system. The expressions depend on the quadrupole moment of the source, as in standard general relativity, but also contain monopolar and dipolar terms. There exists a one-parameter family of Einstein-aether theories for which only the quadrupolar contribution is present, and for which the expression for the damping rate is identical to that of general relativity to the order worked to here. This family cannot yet be declared observationally viable, since effects due to the strong internal fields of bodies in the actual systems used to test the damping rate are not included.

This work concerns the loss of energy of a material system due to gravitational radiation in Einstein-aether theory-an alternative theory of gravity in which the metric couples to a dynamical, timelike, unit-norm vector field. Derived to lowest post-Newtonian order are waveforms for the metric and vector fields far from a nearly Newtonian system and the rate of energy radiated by the system. The expressions depend on the quadrupole moment of the source, as in standard general relativity, but also contain monopolar and dipolar terms. There exists a one-parameter family of Einstein-aether theories for which only the quadrupolar contribution is present, and for which the expression for the damping rate is identical to that of general relativity to the order worked to here. This family cannot yet be declared observationally viable, since effects due to the strong internal fields of bodies in the actual systems used to test the damping rate are not included.

We show that any Lorentz violating theory with two or more propagation speeds is in conflict with the generalized second law of black hole thermodynamics. We do this by identifying a classical energy-extraction method, analogous to the Penrose process, which would decrease the black hole entropy. Although the usual definitions of black hole entropy are ambiguous in this context, we require only very mild assumptions about its dependence on the mass. This extends the result found by Dubovsky and Sibiryakov, which uses the Hawking effect and applies only if the fields with different propagation speeds interact just through gravity. We also point out instabilities that could interfere with their black hole perpetuum mobile, but argue that these can be neglected if the black hole mass is sufficiently large.