on scales usable for our experiment. If we assume for the sake of curiosity that we could construct a cable with the same dimensions as before out of these colossal carbon tubes, we could theoretically spin it up to more than 20 kHz - it would still only radiate 10−32W in gravitational radiation.
This highlights the complete infeasability of producing any measurable amount of GW power in such a laboratory experiment any time in the foreseeable future.
2.2 a short review of general relativity
So far, we have only computed the amount of energy radiated by GWs. Now, we want to sketch how they interact with matter, and how they can be detected directly. We will first review the basic mathematical toolset used to describe GR, give the Einstein field equations as well as their approximate wave solution, and finally discuss the principle behind the direct GW detections up to this date.
All material presented in this section is adapted from the literature, in partic-ular using [37], [73] and [58]. See there for a much more detailed treatment, as well as proofs which are omitted here.
2.2.1 Introduction
A guiding principle in modern physics is that of general covariance. Simply put, it means that our description of physics should not depend on our choice of coordinates. In Special Relativity, this principle is only fulfilled for coordinate transformations between inertialreference frames. Extending it to allow invariance of the physical laws underanydifferentiable coordinate transformation ultimately lead to the discovery of the general theory of relativity.
Contrary to classical mechanics - where space is seen as isomorphic to the three dimensional flat space with time as an absolute parameter governing the laws of physics - space and time are seen as coordinates of a4-dimensional space-timeM. The points of this spacetime are calledevents- they uniquely identify a time and place.
Mathematically,Mcan be described using the tool-set of pseudo-Riemannian geometry. The central object in this theory is a so called manifold, which can
be There are some
additional technical restricitions these have to fulfill, which are not relevant here.
any set of points,M, which are locally equivalent to pseudo-Euclidean space. Formally, this meansM must be equipped with a set of local maps x:M →R4which give a one-to-one correspondence of the area around any pointr ∈ Mto the usual vector spaceR4. These allow us to at least locally -define coordinates in the familiarR4 to mathematically describe physics in a curved space. We consider only differantiable manifolds, for which any
12 gravitational wave astronomy
two maps x, y around the same point p are compatible in such a way that x◦y−1:R4 →R4is infinitely differentiable.
The coordinates of any spacetime event p in a given map x are therefore specified by four numbers, which we indicate byx0(p), x1(p), x2(p),x3(p). We will use greek indices to denote any of these4coordinates. Latin indices denote just the spatial components x1(p),x2(p),x3(p), while the temporal coordinate is identified withx0(p). Furthermore, we will usually drop the explicit dependence on the spacetime eventp, and just write the coordinates asxµ.
Practically, this abstract structure of a manifold equipped with maps allows us to define quantities onM itself, which are independent of the choice of coordinates, thus fullfilling the principle of general covariance.
2.2.2 Vectors, co-vectors and tensors
One common example of such a quantity is a vector field. Given any coordi-nate chartx around a pointp, we can define a basis of thetangent space TpM in that point as tangent space can then be written as
v(p) =
where we introduce the Einstein sum convention - repeated identical indices of upper and lower indices are to be summed. A vector field is then
TM(without the subscriptp) is the tangent bundle. It can be defined as the set of all vector fields onM.
a map v:M → TM, which assigns a vector in TpM for each point, p, in the manifold. Since our manifold represents4 dimensional space-time, we call members of its tangent space4-vectors, to distinguish them from the3 component vectors used in classical physics.
Another way to look at vectors in the point p is to interpret them as the derivative along a curveγ:R→ M withγ(0) = p. Such a curve could, for example, describe the trajectory of a particle moving through spacetime. In that case, it is also called a world line of that particle.
Given a
differentiable function f :M → R, we can define its derivative along the curveγin the point pas
2.2 a short review of general relativity 13
where the last equality See [37] for a detailed
derivation.
relates to the previous definition of a vector using the definitions
If the curveγdescribes the world line of a particle parametrized by its Theproper timeof a particle is the time shown by a perfect clock comoving with it.
proper time, it’s derivative ˙γ with components ˙γµ is called the 4-velocity of the particle.
Closely related to the tangent space is the co-tangent space Tp∗M, which is defined as the space of linear functionsη: TpM → R, or dual space ofTpM.
where δµν is the Kronecker delta. Consequently, we can define a general co-vector in the point pas
k(p) =kµ(p)dxµ
p . (2.19)
Similar to a vector field, a co-vector field is then Similarly toTM,T∗M is theCotangent bundle, the set of all co-vector fields on M.
a mapk:M → T∗Mwhich assigns a co-vector in Tp∗Mto each point p of the manifold. In the following, we will drop the explicit mention of the eventp for vectors, co-vectors and the more general tensors defined below.
A co-vector’s action on a vectorvcan simply be computed as
k(v) =kµvµ, (2.20)
to be evaluated for each point of the manifold.
We can use vector and co-vector fields to construct more general fields of
(m,n) tensorsT∈ TmnM, Employing the sum
which can be written as T=Tαβ11...α...βmn ∂
∂xβ1 ⊗. . .∂x∂βn ⊗dxα1 ⊗. . . dxαm . (2.21) Such a (m,n) tensor can be thought of as a function attached to each point of the manifold which mapsmvectorsv1, . . . ,vm andnco-vectors k1, . . . ,kn to one real number:
T(k1, . . . ,kn,v1, . . . ,vm) =Tαβ11...α...βmn·k1β1. . .knβn·vα11. . .vαmm. (2.22)
2.2.3 The metric tensor
2.2.3.1 Measuring distances
By definition, each pseudo-Riemannian manifold is equipped with ametric tensor g ∈T20M, defining a scalar product at each point. Physically, it can be
14 gravitational wave astronomy
used to define distances between events in the curved spacetime. As described above, since it is a tensor, we can always write its action on two vectorsv,w as
g(v,w) = gµνvµwν, (2.23)
such that it is convenient to write its coefficients,gµν, as a matrix.
The two vectors could represent the derivative, ˙γ(λ),
Contrary to massive
along the worldline of a photon. Photons follow null-geodesics3, fulfilling
g(γ˙(λ), ˙γ(λ)) =0 ∀λ. (2.24)
One important example of a metric is the Minkowski metric tensor,ηµν, of special relativity, which can be written globally in an inertial frame as
η=
This means that in an inertial frame in a flat spacetime, eq. (2.24) simply becomes
˙
γ0(λ)2= δijγ˙i(λ)γ˙j(λ), (2.26) where we used the definitions in eqs. (2.16), (2.17a) and (2.18).
We can integrate the square root of this expression between two events Aand Bto represent a macroscopic spacetime interval:
∆t = The left-hand side is then the
Note that while
∆tand∆Lwill depend on the chosen reference frame.
coordinate time difference between the events, while the right-hand side is the spatial distance between them. This means we can measure spatial distances by tracking the time of flight of photons, which is the basic principle behind interferometric distance measurements.
2.2.3.2 Converting vectors to co-vectors
We can use a metric to convert a vectorvinto a co-vectorv[by defining v[ =g(v,·) =gµνvµ
| {z }
=vν
dxν . (2.28)
Conversely, we can use the dual of the metric, defined via
gµα(g−1)αν=δνµ, (2.29)
3 This is an additional property in addition to the geodesic equation, see section2.2.4below.
2.2 a short review of general relativity 15
These rules can directly be generalized to pull any numer of indices of a general tensor up or down, by applying the same rule for each index.
Note that using eqs. (2.29) and (2.30), we get
gµν = (g−1)µα(g−1)νβgαβ = (g−1)µν. (2.31) This means the components ofg−1 are exactly given by those of gwith the indices pulled up, such that we will use the same symbol for gandg−1.
2.2.4 Derivatives and geodesics
We need a structure on the spacetime manifold to allow us to take derivatives of tensor fields. There are different ways to define such a derivative, but the most prevelant one used in GR is thecovariant derivative or more general a
We consider here only a special case called theLevi-Civita connection.
connection. The covariant derivative ∇of a general tensorT ∈TmnMalong a vectorv= vµ ∂∂xµ can be defined as
The symbolsΓµνκare called the Christoffel symbols (or more generally, the con-nection components). For the Levi-Civita concon-nection, they can be computed from the metric coefficients:
We can use the covariant derivative to define a condition for free-falling particles. A free-falling particle’s world line is a curve γ which naturally follows the background spacetimes curvature. In other words, the
As mentioned above, be used in the geodesic equation
instead.
particles 4-velocity does not change along this trajectory, meaning that there is no acceleration acting on it.
Formally, this means that
∇γ˙γ˙ =0 . (2.35)
Such a curve is called ageodesic.
16 gravitational wave astronomy
2.2.5 Curvature and the Einstein field equations
The metric alone is not sufficient to judge if spacetime is curved at a particular point or not. Instead, one has to compute the Riemann curvature tensor Riem∈ T31M, whose components in a given basis are
Riemµναβ= ∂
∂xαΓµβν−∂x∂βΓµαν+ΓµαγΓγβν−ΓµβγΓγαν. (2.36) A spacetimeM is considered flat if Riem=0 globally. The contraction
Ricµν =Riemγµγν (2.37)
is called the Ricci tensor, and it’s contraction
R=Ricµµ (2.38)
is called the Ricci scalar.
Utilizing these quantities, we can write down theEinstein field equations, Ricµν−
10fundamental equations which relate the spacetime geometry to the matter content encoded in the stress-energy tensorTµν, which appears scaled by the Einstein gravitational constant κ. Λ is a free parameter, the cosmological constant, which can be related to the rate of expansion of the universe.
Note that this is a system of non-linear differential equations for the spacetime metricgµν, since the curvature terms Ricµνand Rcontain derivatives of the Christoffel symbolsΓαµν, which themselves contain derivatives of the metric gµν.
Note that in order to conserve general covariance, any field equation for the metric has to leave4of it’s10degrees of freedom un-constraint.
2.2.6 Gravitational waves as weak-field solutions
Solving the full Einstein field equations is a significant challenge. This remains true even when considering special cases, such as vacuum solutions satisfying Tµν =0.
Exact analytical solutions do exist, for example the famous Schwarzschild met-ric describing a non-rotating sphemet-rical black hole, but are relatively rare. In addition, it is possible to numerically solve the equations, which allows accu-rate modelling of the gravitational waveforms geneaccu-rated by events involving strong gravitational fields, such as the merging of two black holes.
For describing the action of GWs far away from their generating source, however, it has proven more succesful to consider them as a
This decomposition is not valid under arbitrary coordinate transformation, but only under a subset of so-calledgauge transformations, see [58].
small pertubation hµν of the metric tensor on a flat background spacetime:
gµν= ηµν+hµν, (2.40)
2.2 a short review of general relativity 17 whereηµν is the Minkowski tensor of special relativity. As shown in e.g. [58], one can define a special coordinate system called the transverse-traceless (TT) gauge in which thehµν fulfill
hµ0 =0, ∂
∂xjhkj =0,
∑
3k=1
hkk =0 . (2.41)
In this case, it can be shown that the vacuum Einstein field equations simplify significantly to just ordinary wave equations,
hkj =0 , (2.42)
whereis the D’Alambertian operator of special relativity.
These linearized equations are identical in form to the usual wave equations appearing in, for example, electromagnetism. Therefore, eq. (2.42) allows simple monochromatic plane-wave solutions. Due to the additional gauge constraints given in eq. (2.41), the solutions of eq. (2.42) have only two degrees of freedom.
For example, we can consider a plane wave propagating in the x3direction.
This solution is given in the TT gauge by
hµν =cos ω(x0−x3)·
with h+ and h× describing the amplitude of the so-called
In GR, these are the only two valid GW polarizations. Some alternative
gravitational theories allow up to6different polarizations, see e.g.
[102].
plus and cross polarizations of the gravitational wave.
The full metric is therefore given by
gµν =
For the moment, we focus on just theh+polarization. If we remember that the metric determines distances between spacetime events, we see that a h+
polarized GW periodically stretches and contracts the x1 direction, while having an equal but opposite effect on the x2 direction.
In principle, this distortion the effect of a passing GW using mechanical
can be measured by two observers A and B, which, in the absence of a GW, are at rest in the given coordinate system and seperated by the flat-spacetime distanceL, both equipped with a perfect clock.
Acan encode the time shown by his clock on an electromagnetic signal, for example by modulating a laser beam, and send it to B. On reception,B can then recover the encoded time, and compare it to the time shown by his own clock to determine the coordinate time difference between the event the beam was emitted and the event he received it.
18 gravitational wave astronomy
As we saw in section2.2.3.1, in the absence of GWs, this time difference will directly yield the unperturbed spatial distance, L.
In a first approximation, assuming that the photon path connectingAandBis aligned with thex1axis and that Lis small compared to the GW wavelength, the GW will cause a change∆Lin the propagation time proportional to the undisturbed separation L[73],
∆L≈h+cos ω(x0−x3)L. (2.45)
B will therefore see a periodic variation in the emission times of the signals received fromA, from which they can deduce the properties of the GW. The same principle can be applied to measure h×, which acts similarly on the diagonal betweenx1 andx2.
Although conceptually simple, such a light-time GW detector is difficult to realize in practice. Typical GW amplitudes observable on earth are of the orderh+/×<10−20, such that both observersAandBneed extremely precise clocks4to actually be able to distinguish the small fluctuations∆Lcaused by GWs from the intrinsic imperfections of their reference clocks.