All of the tests of general relativity to date have confirmed Einstein's theory. The most important test was the slow orbital decay, or inspiral, of two neutron stars — compact stars with masses near that of the sun but confined to only a few miles across — locked in a binary orbit in our galaxy. The measured rate of the inspiral agrees closely with the prediction of general relativity, resulting in a Nobel Prize in for the two physicists who discovered the binary star and made the measurements.
The aLIGO will look for the predicted gravitational waves from colliding black holes and neutron stars, where gravity becomes very strong. It was put into operation about one month ago at two sites in the U. The whole scientific community awaits the outcome with bated breath. What questions have yet to be answered surrounding general relativity?
News Bureau | ILLINOIS
Do you foresee it continuing to shape science as we know it? A few outstanding questions are: Does general relativity correctly predict the exotic properties of black holes and of gravitational waves? We are still awaiting a complete theory that unites gravity with quantum mechanics.
Computation of these variables at a point requires knowledge of the metric perturbation h ab everywhere. This non-locality obscures the fact that the physical, non-radiative degrees of freedom are causal, a fact which is explicit in Lorentz gauge 4. For example, the Riemann tensor components R itjt , which are directly observable by detectors such as LIGO, are given in terms of the gauge-invariant variables as. In the previous subsection we described a splitting of metric perturbations into radiative, non-radiative and gauge pieces.
This splitting requires that the linearized Einstein equations be valid throughout the spacetime. However, this assumption is not valid in the real Universe: many sources of GWs are intrinsically strong field sources and cannot be described using linearized theory, and on cosmological scales the metric of our Universe is not close to the Minkowski metric.
Furthermore, the splitting requires a knowledge of the metric throughout all of spacetime, whereas any measurements or observations can probe only finite regions of spacetime. For these reasons it is useful to consider linearized perturbation theory in finite regions of spacetime, and to try to define gravitational radiation in this more general context. Consider therefore a finite volume in space. Can we split up the metric perturbation h ab in into radiative and non-radiative pieces? In general, the answer is no: within any finite region, GWs cannot be distinguished from time-varying near-zone fields generated by sources outside that region.
One way to see this is to note that in finite regions of space, the decomposition of the metric into various pieces becomes non-unique, as does the decomposition of vectors into transverse and longitudinal pieces. Alternatively, we note that within any finite vacuum region , one can always find a gauge which is locally TT, that is, a gauge which satisfies the conditions 2.
In particular, this applies to the static Coulomb-type field of a point source, as long as the source itself is outside of. Consequently, isolating the TT piece of the metric perturbation does not yield just the radiative degrees of freedom within a local region—a TT metric perturbation may also contain, for example, Coulomb-type fields. Within finite regions of space, therefore, GWs cannot be defined at a fundamental level—one simply has time-varying gravitational fields.
However, there is a certain limit in which GWs can be approximately defined in local regions, namely the limit in which the wavelength of the waves is much smaller than length and timescales characterizing the background metric. As discussed in that section, this limit will always be valid when one is sufficiently far from all radiating sources. The usual notion of 'gravitational force' disappears in general relativity, replaced instead by the idea that freely falling bodies follow geodesics in spacetime. Given a spacetime metric g ab and a set of spacetime coordinates x a , geodesic trajectories are given by the equation.
By writing the derivatives in the geodesic equation 3. Let us now specialize to linearized theory, with the non-flat part of our metric dominated by a GW in TT gauge. Further, let us specialize to non-relativistic motion for our test body. This implies that v i 1, and to a good approximation we can neglect the velocity-dependent terms in equation 3.
Does this result mean that the GW has no effect? Certainly not! It just tells us that, in TT gauge the coordinate location of a slowly moving, freely falling body is unaffected by the GW. In essence, the coordinates move with the waves. This result illustrates why, in general relativity, it is important to focus upon coordinate-invariant observables—a naive interpretation of the above result would be that freely falling bodies are not influenced by GWs. In fact, the GWs cause the proper separation between two freely falling particles to oscillate, even if the coordinate separation is constant.
The proper distance L between the two particles in the presence of the GW is given by. Notice that we use the fact that the coordinate location of each particle is fixed in TT gauge! In a gauge in which the particles move with respect to the coordinates, the limits of integration would have to vary. Equation 3. Although we used TT gauge to perform this calculation, the result is gauge-independent; we will derive it in a different gauge momentarily. Notice that h TT xx acts as a strain—a fractional length change. The magnitude h of a wave is often referred to as the 'wave strain'.
The proper distance we have calculated here is a particularly important quantity since it directly relates to the accumulated phase which is measured by laser interferometric GW observatories cf the contribution by Danzmann in this volume. We now give a different derivation of the fractional length change 3. We can regard the coordinate displacement L a as a vector on the first geodesic; this is valid to first order in. Spacetime curvature causes the separation vector to change with time—the geodesics will move further apart or closer together, with an acceleration given by the geodesic deviation equation.
This equation is valid to linear order in L a ; fractional corrections to this equation will scale as , where is the lengthscale over which the curvature varies. A convenient coordinate system for analysing the geodesic deviation equation 3. This coordinate system is defined by the requirements. Here is the radius of curvature of spacetime, given by. It also follows from the gauge conditions 3.
From the metric 3. Notice that —the wave's curvature scale is much larger than the lengthscale characterizing its variations. Therefore, we can simply identify L as the proper separation. We now evaluate the geodesic deviation equation 3. From the conditions 3. Note that the key quantity entering into the equation, R itjt , is gauge-invariant in linearized theory, so we can use any convenient coordinate system to evaluate it.
Using the expression 2. This equation is ideal for analysing an interferometric GW detector. We choose Cartesian coordinates such that the interferometer's two arms lie along the x - and y -axes, with the beam splitter at the origin. For concreteness, let us imagine that the GW propagates along the z -axis. From equation 3. Here the subscripts x and y denote the two different arms, not the components of a vector. These distance changes are then measured via laser interferometry. Notice that the GW which is typically a sinusoidally varying function acts tidally, squeezing along one axis and stretching along the other.
Of course, we do not expect nature to provide GWs that so perfectly align with our detectors. In general, we will need to account for the detector's antenna pattern, meaning that we will be sensitive to some weighted combination of the two polarizations, with the weights depending upon the location of a source on the sky, and the relative orientation of the source and the detector. Finally, in our analysis so far of detection, we have assumed that the only contribution to the metric perturbation is the GW contribution. However, in reality time-varying near-zone gravitational fields produced by sources in the vicinity of the detector will also be present.
From the general expression 2. Gravitational waves are generated by the matter source term on the right-hand side of the linearized Einstein equation. In this section we will compute the leading-order contribution to the spatial components of the metric perturbation for a source whose internal motions are slow compared to the speed of light 'slow-motion sources'. We will then compute the TT piece of the metric perturbation to obtain the standard quadrupole formula for the emitted radiation. Equation 4. A wave equation with source generically takes the form.
The Green's function G t , x ; t ', x ' is the field which arises due to a delta function source; it tells how much field is generated at the 'field point' t , x per unit source at the 'source point' t ', x ' :. The field which arises from our actual source is then given by integrating the Green's function against s t , x :. The Green's function associated with the wave operator is very well known see, e. The speed of light c has been restored here to emphasize the causal nature of this Green's function; we set it back to unity in what follows.
As already mentioned, the radiative degrees of freedom are contained entirely in the spatial part of the metric, projected transverse and traceless. Firstly, consider the spatial part of the metric:. We have raised indices on the right-hand side, using the rule that the position of spatial indices in linearized theory is irrelevant.
We now evaluate this quantity at large distances from the source. We also make the same replacement in the time argument of T ij :. These replacements give. To get the rest of the way there, we need to massage this equation a bit. Breaking this up into time and space components, we have. Multiply both sides of this equation by x i x j. We first manipulate the left-hand side:.
Next, manipulate the right-hand side of equation 4. We thus have. In going from the first line to the second, we used the fact that the second and third terms under the integral are divergences. Using Gauss's theorem, they can thus be recast as surface integrals; taking the surface outside the source, their contribution is zero. In going from the second line to the third, we used the fact that the integration domain is not time-dependent, so we can take the derivatives out of the integral. Defining the second moment I ij of the mass distribution via. When we subtract the trace from I ij , we obtain the quadrupole moment tensor:.
To complete the derivation, we must project out the non-TT pieces of the right-hand side of equation 4. It is useful to introduce the projection tensor,. This tensor eliminates vector components parallel to n , leaving only transverse components. Substituting equation 4. Our derivation of the quadrupole formula 4.
In particular, the derivation is not applicable to systems with weak Newtonian gravity whose dynamics are dominated by self-gravity, such as binary star systems 7. This shortcoming of the above linearized-gravity derivation of the quadrupole formula was first pointed out by Eddington. However, it is very straightforward to extend the derivation to encompass systems with non-negligible self-gravity.
In full general relativity, we define the quantity via. When gravity is weak this definition coincides with our previous definition of as a trace-reversed metric perturbation. We impose the harmonic gauge condition. Taking a coordinate divergence of this equation and using the gauge condition 4.
Equations 4. Therefore, the derivation of the last subsection carries over, with the modification that the formula 4. In this equation the term t tt describes gravitational-binding energy, roughly speaking. For systems with weak gravity, this term is negligible in comparison with the term T tt describing the rest-masses of the bodies. Therefore, the quadrupole formula 4. The rough form of the leading GW field that we just derived, equation 4. First, we define some moments of the mass distribution. The zeroth moment is just the mass itself:. More accurately, this is the total mass-energy of the source.
Next, we define the dipole moment:. L i is a vector with the dimension of length; it describes the displacement of the centre of mass from our chosen origin. As such, M 1 is clearly not a very meaningful quantity—we can change its value simply by choosing a different origin. The first moment is the spin angular momentum:.
Using dimensional analysis and simple physical arguments, it is simple to see that the first moment that can contribute to GW emission is M 2. Consider first M 0. We want to combine M 0 with the distance to our source, r , in such a way as to produce a dimensionless wavestrain h. Does this formula make sense for radiation? Not at all!
Conservation of mass-energy tells us that M 0 for an isolated source cannot vary dynamically. This h cannot be radiative; it corresponds to a Newtonian potential, rather than a GW. How about the moment M 1? In order to get the dimensions right, we must take one time derivative:. The extra factor of c converts the dimension of the time derivative to space, so that the whole expression is dimensionless.
Think carefully about the derivative of M 1 :. This is the total momentum of our source. Our guess for the form of a wave corresponding to M 1 becomes. Can this describe a GW? Again, not a chance: the momentum of an isolated source must be conserved. Terms like this can only be gauge artifacts; they do not correspond to radiation. Indeed, terms like 4. How about S 1? Dimensional analysis tells us that radiation from S 1 must take the form. Conservation of angular momentum tells us that the total spin of an isolated system cannot change, so we reject this term for the same reason that we rejected 4.
There is no conservation principle that allows us to reject this term. Comparing to equation 4. In order to generate interesting amounts of GWs, the quadrupole moment's variation must be enormous. The only interesting sources of GWs will be those which have very large masses undergoing extremely rapid variation; even in this case, the strain we expect from typical sources is tiny. The smallness of GWs reflects the fact that gravity is the weakest of the fundamental interactions. Consider a binary star system, with stars of mass m 1 and m 2 in a circular orbit with separation R.
The quadrupole moment is given by. Let us further choose to evaluate the field on the z -axis, so that n points in the z -direction. The projection operators in equation 4. Bearing this in mind, the quadrupole formula 4. The magnitude h of a typical non-zero component of h TT ij is. Such binaries are so common that they are likely to be a confusion-limited source of GWs for space-based detectors, acting in some cases as an effective source of noise. The second line contains typical parameter values for binary neutron stars that are on the verge of spiralling together and merging.
Such waves are targets for the ground-based detectors that have recently begun operations. The tiny magnitude of these waves illustrates why detecting GWs is so difficult. At the most fundamental level, GWs can only be defined within the context of an approximation in which the wavelength of the waves is much smaller than lengthscales characterizing the background spacetime in which the waves propagate.
In this section, we discuss perturbation theory of curved spacetimes, describe the approximation in which GWs can be defined, and derive the effective stress tensor which describes the energy content of GWs. Throughout this section we will for simplicity restrict attention to vacuum spacetime regions. Here we allow g ab B to be any vacuum solution of the Einstein equations. The derivation of the linearized Einstein equation proceeds as before. Some of the formulae acquire extra terms involving coupling to the background Riemann tensor.
Inserting equation 5. The result 5. Next, insert the expansion 5. Dropping the primes, the metric perturbation is thus traceless and transverse:. In this gauge, the linearized Einstein equation 5. Since the quantity 5. The wave equation 5. In the limit discussed below where the wavelength of the waves is much smaller than the lengthscales characterizing the background metric, these couplings have the effect of causing gradual evolution in the properties of the wave.
These gradual changes can be described using the formalism of geometric optics, which shows that GWs travel along null geodesics with slowly evolving amplitudes and polarizations. Outside the geometric optics limit, the curvature couplings in equation 5. The linear perturbation formalism described in the last section can be applied to any perturbation of any vacuum background spacetime. Its starting point is the separation of the spacetime metric into a background piece plus a perturbation.
In most circumstances, this separation is merely a mathematical device and can be chosen arbitrarily; no unique separation is determined by local physical measurements. In this case, one can define the background metric and perturbation, to linear order, via.
Here the angular brackets A useful analogy to consider is the surface of an orange, which contains curvatures on two different lengthscales: An overall, roughly spherical background curvature analogous to the background metric , and a dimpled texture on small scales analogous to the GW. The regime is called the geometric optics regime. This effective stress tensor contributes to the curvature of the background metric g ab B. This contribution to the curvature is. It follows that , or. Consider now the splitting of the Riemann tensor into a background piece plus a perturbation given by equation 5.
By the definition 5. This Riemann tensor perturbation is often called the GW Riemann tensor ; it is a tensor characterizing the GWs that propagate in the background metric g ab B. Then we have. Thus, the gravitational waveforms seen by observers performing local experiments will just be given by components of the GW Riemann tensor in the observer's local proper reference frames. We remark that the splitting of the metric into a background plus a linear perturbation can sometimes be uniquely defined even in the regime. Some examples are when the background spacetime is static e.
Friedman—Robertson—Walker cosmological models. In these cases the dynamic metric perturbation are not actually GWs, although their evolution can be computed using the linearized Einstein equation. For example, consider the evolution of a metric perturbation mode which is parametrically amplified during inflation in the early Universe. Any excitation of the mode is locally measurable although such modes are usually assumed to start in their vacuum state.
As inflation proceeds, the mode's wavelength redshifts and becomes larger due to the rapid expansion of the Universe, and eventually becomes larger than the Hubble scale ; the mode is then 'outside the horizon'. At this point, excitations in the mode are not locally measurable and are thus not GWs. Finally, after inflation ends, the mode 're-enters the horizon' and excitations of the mode are locally measurable.
The mode is now a true GW once again. However, far from sources of GWs the regime relevant to observations , the two definitions do coincide. We start by discussing the second-order perturbation theory. By inserting the expansion 5. Here G ab [ g cd B ] is the Einstein tensor of the background metric, and G ab 1 [ The explicit expression for G ab 1 [ h cd , g ef B ] is given by equation 5.
It is worth recalling that j ab is a second-order metric perturbation. We must take the calculation to second order to compute the effective stress—energy tensor of the waves, since an averaging is involved—the first-order contribution vanishes by the oscillatory nature of the waves. We now specialize to the geometric optics regime. We split the second-order metric perturbation into a piece j ab that is slowly varying, and a piece. Consider next the average of the second-order Einstein equation 5.
Using the fact that the averaging operation Subtracting equation 5. Equation 5. In the effective Einstein equation 5. The left-hand side is the Einstein tensor of the slowly varying piece of the metric. The right-hand side is the effective stress—energy tensor, obtained by taking an average of the quadratic piece of the second-order Einstein tensor. It follows from equation 5. The effect of the GWs is thus to give rise to a correction j ab to the background metric.
However, any measurements that probe only the long-lengthscale structure of the metric e. Thus, when one restricts attention to long lengthscales, GWs can thus be treated as any other form of matter source in general relativity. A fairly simple expression for the effective stress—energy tensor can be obtained as follows. Schematically, the effective stress—energy tensor has the form.
Gravitational fields and the theory of general relativity
However, the commutator of two derivatives scales as the background Riemann tensor, which is of order. Therefore, up to corrections of order which can be neglected, one can freely commute covariant derivatives in the expression 5. At this stage, we need to discuss which modes can be produced during the two black holes merger that led to the gravitational wave observed by the LIGO collaboration.
The LIGO collaboration estimates that the gravitational wave GW is produced by the coalescence of two black holes: the black holes follow an inspiral orbit before merging, and subsequently going through a final black hole ringdown.
The Secret History of Gravitational Waves
Over 0. In this short paper we have investigated quantum gravitational effects in gravitational waves using conservative effective theory methods which are model independent. We found that quantum gravity leads to new poles in the propagator of the graviton besides the usual massless pole. These new states are massive and couple gravitationally to matter. If kinematically allowed, they would thus be produced in roughly the same amount as the usual massless mode in energetic astrophysical events. A sizable amount of the energy produced in astrophysical events could thus be carried away by massive modes which would decay and lead to a damping of this component of the gravitational wave.
While our back-of-the-envelope calculation indicates that the energy released in the merger recently observed by LIGO was unlikely to be high enough to produce such modes, one should be careful in extrapolating the amount of energy of astrophysical events from the energy of the gravitational wave observed on earth. Skip to main content Skip to sections.
Advertisement Hide. Download PDF. Gravitational Waves in Effective Quantum Gravity. Open Access. First Online: 29 July This process is experimental and the keywords may be updated as the learning algorithm improves. An infinite series of vacuum polarization diagrams contributing to the graviton propagator can be resummed in the large N limit. The position of the complex pole depends on the number of fields in the model. We shall not investigate this Lee-Wick pole further and assume that this potential problem is cured by strong gravitational interactions. The renormalization scale needs to be adjusted to match the number of particles included in the model.
Since the complex poles couple with the same coupling to matter as the usual massless graviton, we can think of them as a massive graviton although strictly speaking these objects have two polarizations only in contrast to massive gravitons that have five. This idea has been applied in the context of F R gravity [ 14 ] see also [ 15 , 16 ] for earlier works on gravitational waves in F R gravity. We shall assume that these massive modes can be excited during the merger of two black holes. As a rough approximation, we shall assume that all the energy released during the merger is emitted into these modes.
Given this assumption, we can use the limit derived by the LIGO collaboration on a graviton mass. Abbott, et al. Calmet, Effective theory for quantum gravity. Modern Phys. D 22 , Aydemir, M. Anber, J. Donoghue, Self-healing of unitarity in effective field theories and the onset of new physics. D 86 , Donoghue, B. D 89 10 , Calmet, R.