Chapter I: Introduction
1.3 Gravitational waves
An interesting consequence of general relativity is its prediction of gravitational waves. In the previous section, the concept of a spacetime fabric was introduced.
Gravitational waves are perturbations to this spacetime fabric, wrinkles in the spacetime that propagate at the speed of light. As gravitational sources accelerate, the change in the gravitational field propagates through the spacetime, like ripples in a pond. When a gravitational wave passes an observer, the observer’s local gravitational field is modified by these waves; the local metric is perturbed and the
line element, which is a function of the metric (see Eq. (1.1)) and determines how time and distances are measured, is consequently modified.
More formally, far from any sources, the local metricgabis just Minkowski space, or flat space,ηab. However, in the presence of a gravitational wave, the local metric becomes
gab =ηab+hab, (1.4)
where habis the small perturbation to flat space carried by the gravitational waves.
Following [1], after linearizing the Einstein equations and some manipulation, it can be shown that small perturbations behave as
hab = Aab exp(ikcxc), (1.5) which describes a wave propagating at the speed of light (see [5,6], for more detailed discussions of gravitational waves).
Figure 1.4: A simple illustration of how the Laser Interferometer Gravitational-Wave Observatory (LIGO) functions. A high-power laser passes through a beam splitter, sending light down each of the 4 km arms. Large mirrors at the end of each arm reflect the light back to the beam splitter, which redirects some light from each arm onto a photodetector. The photodetector is sensitive to how the light from each arm interacts with each other and allows for the detection of small changes in the arm lengths due to passing gravitational waves. Credit: T. Pyle, Caltech/MIT/LIGO Lab The LIGO detectors were cleverly designed to detect gravitational waves by measuring these small modifications to our local gravitational field. High-powered lasers are directed down the two arms of the L-shaped detector, are reflected back by large mirrors, and are rejoined at a photodetector that monitors how the two beams interact.
If the local metric remains unmodified, the distance that light travels down each arm of the detector remains unchanged and so does the light interference pattern. When a gravitational wave passes through a detector, the distance that light travels down each arm is slightly longer or shorter, so that when the beams of light reconvene at the photodetector, the interference pattern is consequently modified.
However, this change in distance due to gravitational waves is incomprehensibly small. The ratio of the change of distance along one arm,∆L, over the length of the arm,L, is referred to as the strain
h = ∆L
L , (1.6)
which is directly relatable to Eq. (1.5) [4]. Rearranging Eq. (1.6) and writing this as
∆L = h× L, (1.7)
we see that the change in length is directly proportional to the gravitational wave strain amplitude and the length of the arms. Given the knowledge of expected strain amplitudes from astrophysical source, the exceptionally long (4 km) LIGO arms were chosen with Eq. (1.7) in mind – bringing∆L into a measurable regime.
To get an idea of how small the gravitational strain amplitude actually is, we can turn to the quadrupole formula, which was first derived by Einstein in 1918 [7]. The quadrupole formula provides us with a rough approximation for the strain amplitude of waves emitted from two equal mass objects with mass M [M], orbiting at a separationr[M], located at a distanceR[Mpc] from a detector (withG= c=1) [8]:
h∼5×10−20 1 Mpc R
! M M
! M r
!
. (1.8)
Let us assume that our source is two objects, each with the mass of ten suns, and it is located in one of the closest superclusters of galaxies, the Coma Supercluster, which is at a distance of roughly 90 Mpc away from Earth. This yieldsh∼ 5×10−21
M
r
. The minimum separation achievable by the two objects is approximately twice their radii. If we assume the objects are black holes, this gives rise tor = 4M and we get an estimated strain ofh∼ 1×10−21. This corresponds to a change in the length of the LIGO arms of∆L ∼5×10−18m, which is about 100 times smaller than the radius of a proton. Detecting changes in length at this scale is an unimaginable feat, earning LIGO the title of the most precise ruler ever constructed.
If we repeat the above, assuming the same total mass but replacing the source with two stars rather than compact black holes, the stars enormous sizes prevent them
from reaching the separations obtainable by black holes. With the radius of the sun being roughly 2×105larger than a black hole of the same mass, the two stars would produce, at most, a change of ∆L ∼ 2×10−23m in the LIGO detectors – this is smaller than the size of the smallest known quark. This difference in strain amplitude highlights why compact objects are the primary sources of gravitational waves. Although orbiting stars generate gravitational waves, we are currently only sensitive to those coming from the most extreme objects in space. Fortunately, compact objects are arguably some of the most mysterious objects in our universe, and LIGO has granted us the ability to study these strange beasts comfortably from Earth.