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Thank You! The modified terms split naturally into those that are odd under CPT i. The result is a rich and complex framework, with many parameters to be analyzed and tested by experiment. Such details are beyond the scope of this review; for a review of SME and other frameworks, the reader is referred to the Living Review by Mattingly [ ].
Here we confine our attention to the electromagnetic sector, in order to link the SME with the c 2 framework discussed above. It has 19 independent components. In the rest frame of the universe, these tensors have some form that is established by the global nature of the solutions of the overarching theory being used.
In a frame that is moving relative to the universe, the tensors will have components that depend on the velocity of the frame, and on the orientation of the frame relative to that velocity. With this assumption, all the tensorial quantities in Equation 24 vanish in the preferred frame, and, after suitable rescalings of coordinates and fields, the action 22 can be put into the form of the c 2 framework, with. Thus far, we have discussed EEP as a principle that strictly divides the world into metric and non-metric theories, and have implied that a failure of EEP might invalidate metric theories and thus general relativity.
On the other hand, there is mounting theoretical evidence to suggest that EEP is likely to be violated at some level, whether by quantum gravity effects, by effects arising from string theory, or by hitherto undetected interactions. Roughly speaking, in addition to the pure Einsteinian gravitational interaction, which respects EEP, theories such as string theory predict other interactions which do not. In string theory, for example, the existence of such EEP-violating fields is assured, but the theory is not yet mature enough to enable a robust calculation of their strength relative to gravity, or a determination of whether they are long range, like gravity, or short range, like the nuclear and weak interactions, and thus too short range to be detectable.
The gravitational Lagrangian here takes the form of a scalar-tensor theory see Section 3. But the non-metric electromagnetic term will, in general, produce violations of EEP. For examples of specific models, see [ , 85 ]. Unlike the fields of the standard model of electromagnetic, weak and strong interactions, which couple to properties other than mass-energy and are either short range or are strongly screened, the fields inspired by string theory could be long range if they remain massless by virtue of a symmetry, or at best, acquire a very small mass , and can couple to mass-energy, and thus can mimic gravitational fields.
Still, there appears to be no way to make this precise. Whether current or proposed experiments can actually probe these phenomena meaningfully is an open question at the moment, largely because of a dearth of firm theoretical predictions. This proposal dovetailed with earlier hints of a deviation from the inverse-square law of Newtonian gravitation derived from measurements of the gravity profile down deep mines in Australia, and with emerging ideas from particle physics suggesting the possible presence of very low-mass particles with gravitational-strength couplings.
Although two early experiments reported positive evidence, the others all yielded null results. Despite early reports of anomalies, independent tower, borehole, and seawater measurements ultimately showed no evidence of a deviation. Analyses of orbital data from planetary range measurements, lunar laser ranging LLR , and laser tracking of the LAGEOS satellite verified the inverse-square law to parts in 10 8 over scales of 10 3 to 10 5 km, and to parts in 10 9 over planetary scales of several astronomical units [ ].
A consensus emerged that there was no credible experimental evidence for a fifth force of nature, of a type and range proposed by Fischbach et al. For reviews and bibliographies of this episode, see [ , , , 4 , ]. Other possibilities for effective modifications of gravity at short range involved the exchange of light scalar particles. Following these proposals, many of the high-precision, low-noise methods that were developed for tests of WEP were adapted to carry out laboratory tests of the inverse square law of Newtonian gravitation at millimeter scales and below.
The challenge of these experiments has been to distinguish gravitation-like interactions from electromagnetic and quantum mechanical Casimir effects. For a comprehensive review of both the theory and the experiments, see [ 3 ]. The empirical evidence supporting the Einstein equivalence principle, discussed in the previous Section 2 , supports the conclusion that the only theories of gravity that have a hope of being viable are metric theories, or possibly theories that are metric apart from very weak or short-range non-metric couplings as in string theory. Therefore for the remainder of this review, we shall turn our attention exclusively to metric theories of gravity, which assume that.
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Because of it, one can discuss the metric as a property of spacetime itself rather than as a field over spacetime. This is because its properties may be measured and studied using a variety of different experimental devices, composed of different non-gravitational fields and particles, and, because of universal coupling, the results will be independent of the device. Thus, for instance, the proper time between two events is a characteristic of spacetime and of the location of the events, not of the clocks used to measure it.
In any metric theory of gravity, matter and non-gravitational fields respond only to the spacetime metric g. In principle, however, there could exist other gravitational fields besides the metric, such as scalar fields, vector fields, and so on. If, by our strict definition of metric theory, matter does not couple to these fields, what can their role in gravitation theory be? Their role must be that of mediating the manner in which matter and non-gravitational fields generate gravitational fields and produce the metric; once determined, however, the metric alone acts back on the matter in the manner prescribed by EEP.
What distinguishes one metric theory from another, therefore, is the number and kind of gravitational fields it contains in addition to the metric, and the equations that determine the structure and evolution of these fields. In other words, the behavior of each field is influenced to some extent by a coupling to at least one of the other fields in the theory.
Brans-Dicke theory and its generalizations are purely dynamical theories; the field equation for the metric involves the scalar field as well as the matter as source , and that for the scalar field involves the metric. Consider a local, freely falling frame in any metric theory of gravity. Let this frame be small enough that inhomogeneities in the external gravitational fields can be neglected throughout its volume. On the other hand, let the frame be large enough to encompass a system of gravitating matter and its associated gravitational fields.
The system could be a star, a black hole, the solar system, or a Cavendish experiment. To determine the behavior of the system we must calculate the metric. The computation proceeds in two stages. Second, we solve for the fields generated by the local system. But because the metric is coupled directly or indirectly to the other fields of the theory, its structure and evolution will be influenced by those fields, and in particular by the boundary values taken on by those fields far from the local system.
Thus the gravitational environment in which the local gravitating system resides can influence the metric generated by the local system via the boundary values of the auxiliary fields. Consequently, the results of local gravitational experiments may depend on the location and velocity of the frame relative to the external environment.
Of course, local non -gravitational experiments are unaffected since the gravitational fields they generate are assumed to be negligible, and since those experiments couple only to the metric, whose form can always be made locally Minkowskian at a given spacetime event. We can now make several statements about different kinds of metric theories. A theory which contains only the metric g yields local gravitational physics which is independent of the location and velocity of the local system. This follows from the fact that the only field coupling the local system to the environment is g , and it is always possible to find a coordinate system in which g takes the Minkowski form at the boundary between the local system and the external environment neglecting inhomogeneities in the external gravitational field.
General relativity is an example of such a theory.
The Confrontation between General Relativity and Experiment
This follows from the asymptotic Lorentz invariance of the Minkowski metric and of the scalar fields, but now the asymptotic values of the scalar fields may depend on the location of the frame. On the other hand, a form of velocity dependence in local physics can enter indirectly if the asymptotic values of the scalar field vary with time cosmologically. Then the rate of variation of the gravitational constant could depend on the velocity of the frame.
A theory which contains the metric g and additional dynamical vector or tensor fields or prior-geometric fields yields local gravitational physics which may have both location and velocity-dependent effects. These ideas can be summarized in the strong equivalence principle SEP , which states that:. The outcome of any local test experiment is independent of the velocity of the freely falling apparatus. The outcome of any local test experiment is independent of where and when in the universe it is performed.
The distinction between SEP and EEP is the inclusion of bodies with self-gravitational interactions planets, stars and of experiments involving gravitational forces Cavendish experiments, gravimeter measurements. The above discussion of the coupling of auxiliary fields to local gravitating systems indicates that if SEP is strictly valid, there must be one and only one gravitational field in the universe, the metric g. These arguments are only suggestive however, and no rigorous proof of this statement is available at present.
General relativity seems to be the only viable metric theory that embodies SEP completely. In Section 3. Despite the possible existence of long-range gravitational fields in addition to the metric in various metric theories of gravity, the postulates of those theories demand that matter and non-gravitational fields be completely oblivious to them. The only gravitational field that enters the equations of motion is the metric g. The role of the other fields that a theory may contain can only be that of helping to generate the spacetime curvature associated with the metric.
Matter may create these fields, and they plus the matter may generate the metric, but they cannot act back directly on the matter. Matter responds only to the metric. Thus the metric and the equations of motion for matter become the primary entities for calculating observable effects, and all that distinguishes one metric theory from another is the particular way in which matter and possibly other gravitational fields generate the metric. The comparison of metric theories of gravity with each other and with experiment becomes particularly simple when one takes the slow-motion, weak-field limit.
This approximation, known as the post-Newtonian limit, is sufficiently accurate to encompass most solar-system tests that can be performed in the foreseeable future. It turns out that, in this limit, the spacetime metric g predicted by nearly every metric theory of gravity has the same structure. These potentials are constructed from the matter variables see Box 2 in imitation of the Newtonian gravitational potential.
A consistent post-Newtonian limit requires determination of g 00 correct through , g 0 i through , and g ij through for details see TEGP 4. The only way that one metric theory differs from another is in the numerical values of the coefficients that appear in front of the metric potentials.
The parametrized post-Newtonian PPN formalism inserts parameters in place of these coefficients, parameters whose values depend on the theory under study. Under reasonable assumptions about the kinds of potentials that can be present at post-Newtonian order basically only Poisson-like potentials , one finds that ten PPN parameters exhaust the possibilities.
Will [ ] generalized the framework to perfect fluids. The canonical version, with conventions altered to be more in accord with standard textbooks such as [ ], is discussed in detail in TEGP 4 [ ]. Other versions of the PPN formalism have been developed to deal with point masses with charge, fluid with anisotropic stresses, bodies with strong internal gravity, and post-post-Newtonian effects TEGP 4. The framework uses a nearly globally Lorentz coordinate system in which the coordinates are t, x 1 ,x 2 ,x 3. Three-dimensional, Euclidean vector notation is used throughout.
Test bodies:. One of the important applications of the PPN formalism is the comparison and classification of alternative metric theories of gravity. The population of viable theories has fluctuated over the years as new effects and tests have been discovered, largely through the use of the PPN framework, which eliminated many theories thought previously to be viable. The theory population has also fluctuated as new, potentially viable theories have been invented. In this review, we shall focus on GR, the general class of scalar-tensor modifications of it, of which the Jordan-Fierz-Brans-Dicke theory Brans-Dicke, for short is the classic example, and vector-tensor theories.
The reasons are several-fold:. A full compendium of alternative theories circa is given in TEGP 5 [ ]. Few of these could be regarded as well-motivated theories from the point of view, say, of field theory or particle physics. A large number of alternative theories of gravity predict gravitational wave emission substantially different from that of general relativity, in strong disagreement with observations of the binary pulsar see Section 7. Scalar-tensor modifications of GR have become very popular in unification schemes such as string theory, and in cosmological model building.
Because the scalar fields could be massive, the potentials in the post-Newtonian limit could be modified by Yukawa-like terms. Vector-tensor theories have attracted recent attention, in the spirit of the SME see Section 2. The metric g is the sole dynamical field, and the theory contains no arbitrary functions or parameters, apart from the value of the Newtonian coupling constant G , which is measurable in laboratory experiments.
The general procedure for deriving the post-Newtonian limit of metric theories is spelled out in TEGP 5. The Einstein frame is useful for discussing general characteristics of such theories, and for some cosmological applications, while the metric representation is most useful for calculating observable effects. A precisely linear coupling function produces Brans-Dicke theory, with , or. Scalar fields coupled to gravity or matter are also ubiquitous in particle-physics-inspired models of unification, such as string theory [ , , 85 , 82 , 83 ].
In some models, the coupling to matter may lead to violations of EEP, which could be tested or bounded by the experiments described in Section 2. Only if the theory can be cast as a metric theory with a scalar field of infinite range or of range long compared to the scale of the system in question solar system can the PPN framework be strictly applied.
If the mass of the scalar field is sufficiently large that its range is microscopic, then, on solar-system scales, the scalar field is suppressed, and the theory is essentially equivalent to general relativity. The most general action for such theories that is quadratic in derivatives of the vector is given by. The coefficients c i are arbitrary. In addition to having up to four arbitrary parameters, they also left the magnitude of the vector field arbitrary, since it satisfies a linear homogenous vacuum field equation of the form in all such cases studied.
Indeed, this latter fact was one of most serious defects of these theories. Will-Nordtvedt theory see [ ]. The general class of theories was analyzed by Jacobson and collaborators [ , , , 99 , ], motivated in part by [ ]. With the PPN formalism in hand, we are now ready to confront gravitation theories with the results of solar-system experiments.
A light ray or photon which passes the Sun at a distance d is deflected by an angle. TEGP 7. Another, more useful expression gives the change in the relative angular separation between an observed source of light and a nearby reference source as both rays pass near the Sun:. Thus, calculations that purport to derive the full deflection using the equivalence principle alone are incorrect.
Its GR value is unity. The arrows at the top denote anomalously large values from early eclipse expeditions. Hipparcos denotes the optical astrometry satellite, which reached 0. However, the development of radio-interferometery, and later of very-long-baseline radio interferometry VLBI , produced greatly improved determinations of the deflection of light. These techniques now have the capability of measuring angular separations and changes in angles to accuracies better than microarcseconds.
A number of measurements of this kind over the period — yielded an accurate determination of the coefficient. Analysis of observations made by the Hipparcos optical astrometry satellite yielded a test at the level of 0. A VLBI measurement of the deflection of light by Jupiter was reported; the predicted deflection of about microarcseconds was seen with about 50 percent accuracy [ ]. For a ray which passes close to the Sun,. In order to do this accurately however, one must take into account the variations in round-trip travel time due to the orbital motion of the target relative to the Earth.
This is done by using radar-ranging and possibly other data on the target taken when it is far from superior conjunction i. The resulting predicted round-trip travel times in terms of the unknown coefficient are then fit to the measured travel times using the method of least-squares, and an estimate obtained for. The Viking experiment resulted in a 0.
This was made possible by the ability to do Doppler measurements using both X-band MHz and Ka-band MHz radar, thereby significantly reducing the dispersive effects of the solar corona. In addition, the superior conjunction of Cassini was particularly favorable: With the spacecraft at 8.
From the results of the Cassini experiment, we can conclude that the coefficient must be within at most 0. In , Kopeikin [ ] suggested that a measurement of the time delay of light from a quasar as the light passed by the planet Jupiter could be used to measure the speed of the gravitational interaction. On September 8, , Jupiter passed almost in front of a quasar, and Kopeikin and Fomalont made precise measurements of the Shapiro delay with picosecond timing accuracy, and claimed to have measured the correction term to about 20 percent [ , , , ].
However, several authors pointed out that this 1. But if that is the case, then the principle of relativity says that one can view things from the rest frame of Jupiter. Kopeikin gave a number of arguments in opposition to this interpretation [ , , , ]. Current limits on the PPN parameters. The modern value for this discrepancy is 43 arcseconds per century.
A number of ad hoc proposals were made in an attempt to account for this excess, including, among others, the existence of a new planet Vulcan near the Sun, a ring of planetoids, a solar quadrupole moment and a deviation from the inverse-square law of gravitation, but none was successful. General relativity accounted for the anomalous shift in a natural way without disturbing the agreement with other planetary observations.
The predicted advance per orbit , including both relativistic PPN contributions and the Newtonian contribution resulting from a possible solar quadrupole moment, is given by. We have ignored preferred-frame and galaxy-induced contributions to ; these are discussed in TEGP 8. We shall drop this term henceforth. The third term depends upon the solar quadrupole moment J 2. Substituting standard orbital elements and physical constants for Mercury and the Sun we obtain the rate of perihelion shift Open in a separate window , in seconds of arc per century,.
Now, the measured perihelion shift of Mercury is known accurately: After the perturbing effects of the other planets have been accounted for, the excess shift is known to about 0. Analysis of data taken since could improve the accuracy. The next class of solar-system experiments that test relativistic gravitational effects may be called tests of the strong equivalence principle SEP. Among the testable violations of SEP are a violation of the weak equivalence principle for gravitating bodies that leads to perturbations in the Earth-Moon orbit, preferred-location and preferred-frame effects in the locally measured gravitational constant that could produce observable geophysical effects, and possible variations in the gravitational constant over cosmological timescales.
In a pioneering calculation using his early form of the PPN formalism, Nordtvedt [ ] showed that many metric theories of gravity predict that massive bodies violate the weak equivalence principle — that is, fall with different accelerations depending on their gravitational self-energy. Dicke [ ] argued that such an effect would occur in theories with a spatially varying gravitational constant, such as scalar-tensor gravity.
For a spherically symmetric body, the acceleration from rest in an external gravitational potential U has the form. This perturbation in the Earth-Moon orbit leads to a polarization of the orbit that is directed toward the Sun as it moves around the Earth-Moon system, as seen from Earth. This polarization represents a perturbation in the Earth-Moon distance of the form.
Since August , when the first successful acquisition was made of a laser signal reflected from the Apollo 11 retroreflector on the Moon, the LLR experiment has made regular measurements of the round-trip travel times of laser pulses between a network of observatories and the lunar retroreflectors, with accuracies that are at the 50 ps 1 cm level, and that may soon approach 5 ps 1 mm. These measurements are fit using the method of least-squares to a theoretical model for the lunar motion that takes into account perturbations due to the Sun and the other planets, tidal interactions, and post-Newtonian gravitational effects.
The predicted round-trip travel times between retroreflector and telescope also take into account the librations of the Moon, the orientation of the Earth, the location of the observatories, and atmospheric effects on the signal propagation. Numerous ongoing analyses of the data find no evidence, within experimental uncertainty, for the Nordtvedt effect [ , ] for earlier results see [ 95 , , ].
These results represent a limit on a possible violation of WEP for massive bodies of about 1. The resulting bound of 1. In GR, the Nordtvedt effect vanishes; at the level of several centimeters and below, a number of non-null general relativistic effects should be present [ ]. Tests of the Nordtvedt effect for neutron stars have also been carried out using a class of systems known as wide-orbit binary millisecond pulsars WBMSP , which are pulsar-white-dwarf binary systems with small orbital eccentricities.
In the gravitational field of the galaxy, a non-zero Nordtvedt effect can induce an apparent anomalous eccentricity pointed toward the galactic center [ 86 ], which can be bounded using statistical methods, given enough WBMSPs see [ ] for a review and references. Some theories of gravity violate SEP by predicting that the outcomes of local gravitational experiments may depend on the velocity of the laboratory relative to the mean rest frame of the universe preferred-frame effects or on the location of the laboratory relative to a nearby gravitating body preferred-location effects.
Most theories of gravity that violate SEP predict that the locally measured Newtonian gravitational constant may vary with time as the universe evolves. Where G does change with cosmic evolution, its rate of variation should be of the order of the expansion rate of the universe, i. The first type of bound typically comes from LLR measurements, planetary radar-ranging measurements, and pulsar timing data.
The second type comes from studies of the evolution of the Sun, stars and the Earth, big-bang nucleosynthesis, and analyses of ancient eclipse data. Constancy of the gravitational constant. For binary pulsar data, the bounds are dependent upon the theory of gravity in the strong-field regime and on neutron star equation of state. Big-bang nucleosynthesis bounds assume specific form for time dependence of G.
For an initial analysis along these lines, see [ ]. In a similar manner, bounds from helioseismology and big-bang nucleosynthesis BBN assume a model for the evolution of G over the multi-billion year time spans involved. For example, the concordance of predictions for light elements produced around 3 minutes after the big bang with the abundances observed indicate that G then was within 20 percent of G today. According to GR, moving or rotating matter should produce a contribution to the gravitational field that is the analogue of the magnetic field of a moving charge or a magnetic dipole.
In particular, one can view the g 0i part of the PPN metric see Box 2 as an analogue of the vector potential of electrodynamics. In a suitable gauge, and dropping the preferred-frame terms, it can be written. Gravitomagnetism plays a role in a variety of measured relativistic effects involving moving material sources, such as the Earth-Moon system and binary pulsar systems. Nordtvedt [ , ] has argued that, if the gravitomagnetic potential 55 were turned off, then there would be anomalous orbital effects in LLR and binary pulsar data.
Rotation also produces a gravitomagnetic effect, since for a rotating body, , where J is the angular momentum of the body. A set of four superconducting-niobium-coated, spherical quartz gyroscopes were flown in a polar Earth orbit km mean altitude, 0. The accuracy goal of the experiment is about 0. The spacecraft was launched on April 20, , and the mission ended in September , as scheduled, when the remaining liquid helium boiled off.
It is too early to know whether the relativistic precessions were measured in the amount predicted by GR, because an important calibration of the instrument exploits the effect of the aberration of starlight on the pointing of the on-board telescope toward the guide star, and completing this calibration required the full mission data set. In addition, part of the measured effect includes the motion of the guide star relative to distant inertial frames. Final results from the experiment are expected in Another way to look for frame-dragging is to measure the precession of orbital planes of bodies circling a rotating body.
Unfortunately, the two existing LAGEOS satellites are not in appropriately inclined orbits, and no concrete plans exist at present to launch a third satellite in a supplementary orbit. In earlier reports, Ciufolini et al. A gyroscope moving through curved spacetime suffers a precession of its spin axis given by. The predicted precession is about 2 arcseconds per century, an effect first calculated by de Sitter.
This effect has been measured to about 0. For the GPB gyroscopes orbiting the Earth, the precession is 6. From such considerations, there follows [ ] the additional theoretical constraint. Such an equality guarantees the equality of action and reaction and of conservation of momentum, at least in the Newtonian limit. A remarkable planetary test was reported by Bartlett and van Buren [ 22 ]. They noted that current understanding of the structure of the Moon involves an iron-rich, aluminum-poor mantle whose center of mass is offset about 10 km from the center of mass of an aluminum-rich, iron-poor crust.
Improved knowledge of the lunar orbit through LLR, and a better understanding of tidal effects in the Earth-Moon system which also contribute to the secular acceleration through satellite data, severely limit any anomalous secular acceleration, with the resulting limit. According to the PPN formalism, in a theory of gravity that violates conservation of momentum, but that obeys the constraint of Equation 58 , the electrostatic binding energy E e of an atomic nucleus could make a contribution to the ratio of active to passive mass of the form.
Nordtvedt [ ] has examined whether this bound could be improved by considering the asymmetric distribution of ocean water on Earth. Another consequence of a violation of conservation of momentum is a self-acceleration of the center of mass of a binary stellar system, given by. It is planned for launch by ESA in the time frame. A recent contribution in that direction is [ ]. This can occur in a number of situations:. The evolution of the system may be affected by the emission of gravitational radiation.
The 1PN approximation does not contain the effects of gravitational radiation back-reaction. In the expression for the metric given in Box 2 , radiation back-reaction effects do not occur until in g 00 , in , and in g ij. Consequently, in order to describe such systems, one must carry out a solution of the equations substantially beyond 1PN order, sufficient to incorporate the leading radiation damping terms at 2.
In addition, the PPN metric described in Section 3. As such it cannot describe the gravitational waves seen by a detector. Systems like this include the late stage of the inspiral of binary systems of neutron stars or black holes, driven by gravitational radiation damping, prior to a merger and collapse to a final stationary state.
Binary inspiral is one of the leading candidate sources for detection by a world-wide network of laser interferometric gravitational wave observatories nearing completion. A proper description of such systems requires not only equations for the motion of the binary carried to extraordinarily high PN orders at least 3.
Of course, some systems cannot be properly described by any post-Newtonian approximation because their behavior is fundamentally controlled by strong gravity. These include the imploding cores of supernovae, the final merger of two compact objects, the quasinormal-mode vibrations of neutron stars and black holes, the structure of rapidly rotating neutron stars, and so on. Phenomena such as these must be analyzed using different techniques. Another is black hole perturbation theory see [ , , ] for reviews.
When dealing with the motion and gravitational wave generation by orbiting bodies, one finds a remarkable simplification within GR. As long as the bodies are sufficiently well-separated that one can ignore tidal interactions and other effects that depend upon the finite extent of the bodies such as their quadrupole and higher multipole moments , then all aspects of their orbital behavior and gravitational wave generation can be characterized by just two parameters: mass and angular momentum.
Whether their internal structure is highly relativistic, as in black holes or neutron stars, or non-relativistic as in the Earth and Sun, only the mass and angular momentum are needed. Furthermore, both quantities are measurable in principle by examining the external gravitational field of the bodies, and make no reference whatsoever to their interiors. It is a consequence of the SEP, described in Section 3. Consider the motion of a body in a binary system, whose size is small compared to the binary separation.
Surround the body by a region that is large compared to the size of the body, yet small compared to the separation. Because of the general covariance of the theory, one can choose a freely-falling coordinate system which comoves with the body, whose spacetime metric takes the Minkowski form at its outer boundary ignoring tidal effects generated by the companion. There is thus no evidence of the presence of the companion body, and the structure of the chosen body can be obtained using the field equations of GR in this coordinate system. Far from the chosen body, the metric is characterized by the mass and angular momentum assuming that one ignores quadrupole and higher multipole moments of the body as measured far from the body using orbiting test particles and gyroscopes.
A black hole of mass m and a planet of mass m would produce identical spacetimes in this outer region. The geometry of this region surrounding the one body must be matched to the geometry provided by the companion body. These are the equations of motion. As a result the motion of two planets of mass and angular momentum m 1 , m 2 , J 1 , and J 2 is identical to that of two black holes of the same mass and angular momentum again, ignoring tidal effects.
This effacement does not occur in an alternative gravitional theory like scalar-tensor gravity. This can affect the value of G inside the chosen body, alter its internal structure specifically its gravitational binding energy and hence alter its mass. The precise nature of the functions will depend on the body, specifically on its gravitational binding energy, and as a result, the motion and gravitational radiation may depend on the internal structure of each body. For compact bodies such as neutron stars and black holes these internal structure effects could be large; for example, the gravitational binding energy of a neutron star can be 10—20 percent of its total mass.
At 1PN order, the leading manifestation of this phenomenon is the Nordtvedt effect. This is how the study of orbiting systems containing compact objects provides strong-field tests of GR. The motion of bodies and the generation of gravitational radiation are long-standing problems that date back to the first years following the publication of GR, when Einstein calculated the gravitational radiation emitted by a laboratory-scale object using the linearized version of GR, and de Sitter calculated N -body equations of motion for bodies in the 1PN approximation to GR.
Damour [ 70 ] gives a thorough historical and technical review of this subject up to The problem of motion and radiation in GR has received renewed interest since , with proposals for construction of large-scale laser interferometric gravitational wave observatories, such as the LIGO project in the US, VIRGO and GEO in Europe, and TAMA in Japan, and the realization that a leading candidate source of detectable waves would be the inspiral, driven by gravitational radiation damping, of a binary system of compact objects neutron stars or black holes [ 1 , ].
The analysis of signals from such systems will require theoretical predictions from GR that are extremely accurate, well beyond the leading-order prediction of Newtonian or even post-Newtonian gravity for the orbits, and well beyond the leading-order formulae for gravitational waves. This presented a major theoretical challenge: to calculate the motion and radiation of systems of compact objects to very high PN order, a formidable algebraic task, while addressing a number of issues of principle that have historically plagued this subject, sufficiently well to ensure that the results were physically meaningful.
This challenge has been largely met, so that we may soon see a remarkable convergence between observational data and accurate predictions of gravitational theory that could provide new, strong-field tests of GR. Here we give a brief overview of the problem of motion and gravitational radiation in GR. However, this is not the most useful form for actual calculations. Equation 62 is exact, and depends only on the assumption that spacetime can be covered by harmonic coordinates.
For example, to obtain the 1PN equations of motion, two iterations are needed i. At the same time, just as in electromagnetism, the formal integral 64 must be handled differently, depending on whether the field point is in the far zone or the near zone. As a result, there is a danger that the integrals involved in the various expansions will diverge or be ill-defined.
Numerous approaches have been developed to try to handle this difficulty. In the DIRE method, all integrals are finite and convergent. Thus, in order to apply the results to cases where the bodies may be neutron stars or black holes, one relies upon the SEP to argue that, if tidal forces are ignored, and equations are expressed in terms of masses and spins, one can simply extrapolate the results unchanged to the situation where the bodies are ultrarelativistic.
While no general proof of this exists, it has been shown to be valid in specific circumstances, such as at 2PN order in the equations of motion, and for black holes moving in a Newtonian background field [ 70 ]. Methods such as these have resolved most of the issues that led to criticism of the foundations of gravitational radiation theory during the s. Among the results of these approaches are formulae for the equations of motion and gravitational waveform of binary systems of compact objects, carried out to high orders in a PN expansion.
Here we shall only state the key formulae that will be needed for this review. For example, the relative two-body equation of motion has the form. The notation A n PN indicates that the term is relative to the Newtonian term. Explicit and unambiguous formulae for non-spinning bodies through 3. Here we quote only the first PN corrections and the leading radiation-reaction terms at 2. These terms are sufficient to analyze the orbit and evolution of the binary pulsar see Section 5.
Another product is a formula for the gravitational field far from the system, written schematically in the form. The leading term is the so-called quadrupole formula. For a binary system this leads to. For binary systems, explicit formulae for the waveform through 2PN order have been derived see [ 40 ] for a ready-to-use presentation of the waveform for circular orbits; see [ 34 ] for a full review.
If the right-hand side of equation is strictly positive, in which case one says that the two events are timelike separated, or have a timelike interval, then one can find an inertial frame with respect to which the two events have the same spatial position. The straight world line joining the two events corresponds to the time axis of this inertial frame of reference. The proper time would be measured by any clock moving along the straight world line between the two events. Just as an ordinary vector like v has three components, v x , v y , and v z , a 4-vector has four components.
Geometrically the 4-velocity and 4-acceleration correspond, respectively, to the tangent vector and the curvature vector of the world line see Figure 2. If the particle moves slower than light, the tangent, or velocity, vector at each event on the world line points inside the light cone of that event, and the acceleration, or curvature, vector points outside the light cone. If the particle moves with the speed of light, then the tangent vector lies on the light cone at each event on the world line. One can, however, define a so-called affine parameter that satisfies equation with zero on the right-hand side.
For the time being this discussion will be restricted to particles moving with speeds less than light. This requirement is invariant under Lorentz transformations of the form of equations and The tangent vector then points inside the future light cone and is said to be future-directed and timelike see Figure 3. One may if one wishes attach an arrow to the world line to indicate this fact.
One says that the particle moves forward in time. It was pointed out by the Swiss physicist Ernest C. It is possible to interpret these world lines in terms of antiparticles, as will be seen when particles moving in a background electromagnetic field are considered. Equations and , which relate the curvature of the world line to the applied forces, are the same in all inertial frames related by Lorentz transformations. It induces an identical rotation on the 4-acceleration and force 4-vectors.
To say that both of these 4-vectors experience the same generalized rotation or Lorentz transformation is simply to say that the fundamental laws of motion and are the same in all inertial frames related by Lorentz transformations. They also have a natural generalization in the general theory of relativity, which incorporates the effects of gravity. The law of motion may also be expressed as:. This idea is a consequence of special relativity alone. It really comes into its own, however, when one considers relativistic quantum mechanics.
This work goes into increasing the energy E of the particle. Taking the dot product of equation with v gives. However, experience has shown that its introduction serves no useful purpose and may lead to confusion, and it is not used in this article. The invariant quantity is the rest mass m. For that reason it has not been thought necessary to add a subscript or superscript to m to emphasize that it is the rest mass rather than a velocity-dependent quantity. When subscripts are attached to a mass, they indicate the particular particle of which it is the rest mass.
If the applied force F is perpendicular to the velocity v , it follows from equation that the energy E, or, equivalently, the velocity squared v 2 , will be constant, just as in Newtonian mechanics. This will be true, for example, for a particle moving in a purely magnetic field with no electric field present. It then follows from equation that the shape of the orbits of the particle are the same according to the classical and the relativistic equations. However, the rate at which the orbits are traversed differs according to the two theories.
The first term, mc 2 , which remains even when the particle is at rest, is called the rest mass energy. For a single particle, its inclusion in the expression for energy might seem to be a matter of convention: it appears as an arbitrary constant of integration. However, for systems of particles that undergo collisions, its inclusion is essential.
The relativistic law of energy - momentum conservation thus combines and generalizes in one relativistically invariant expression the separate conservation laws of prerelativistic physics: the conservation of mass , the conservation of momentum , and the conservation of energy. In fact, the law of conservation of mass becomes incorporated in the law of conservation of energy and is modified if the amount of energy exchanged is comparable with the rest mass energy of any of the particles.
In such a decay the initial kinetic energy is zero. It is precisely this process that provides the large amount of energy available during nuclear fission , for example, in the spontaneous fission of the uranium isotope. The opposite process occurs in nuclear fusion when two particles fuse to form a particle of smaller total rest mass.