### Testing the Strong-Field Dynamics of General Relativity

**Tjonnie G. F. Li**

[Tjonnie G. F. Li is the recipient of the 2013 Stefano Braccini Thesis Prize administered by the Gravitational Wave International Committee (GWIC) for his PhD thesis “Extracting Physics from Gravitational Waves: Testing the Strong-field Dynamics of General Relativity and Inferring the Large-scale Structure of the Universe” (PDF). His thesis work was carried out at Nikhef - Dutch National Institute for Subatomic Physics, the Netherlands and the Ph.D was awarded by Vrije Universiteit, Amsterdam, the Netherlands.

The Stefano Braccini Thesis Prize was established to honor the memory of a talented gravitational wave physicist whose promising career was cut short. Stefano worked with the French-Italian Virgo project, and contributed to the superattentuator design, to the integration and commissioning of Virgo and to its data analysis efforts. -- 2Physics.com]

**Author: Tjonnie G. F. Li**

**Affiliation: Rubicon Postdoctoral Fellow, LIGO Laboratory, California Institute of Technology, USA.**

**Motion of celestial objects**

Humans have been watching the sky for thousands of years. In early times, humans tracked the motion of the Sun and the Moon to make calendars and to associate it with Earthly events such as tides and seasons. By tracking the motion of celestial objects, the early notion of the orbit of the Sun, the Moon and the planets started to form. Building on earlier work developed by Greek astronomers, Claudius Ptolemy (90–168) introduced an accurate model of the planetary orbits by including the notion of a smaller circular orbit (epicycle) augmenting the primary circular orbit.

During the Renaissance, our knowledge of the sky started to change. Johannes Kepler (1571–1630) introduced three laws that described the planetary orbits as ellipses with the Sun at the focus. Later, Isaac Newton (1642-1727) showed that Kepler’s laws of planetary motion can be derived from a law that not only describes the motion of planets, but also describes how all objects are attracted to each other. Newton’s law of universal gravitation states that all objects “pull” on each other through the gravitational force, and the strength of this force is determined by the masses of the two objects.

Despite the success of Newton’s law of universal gravitation, it could not account for the shift in Mercury’s perihelion, the point in Mercury’s orbit that is closest to the Sun. It was Albert Einstein (1879–1955) who refined Newton’s law of universal gravitation by introducing the general theory of relativity. Einstein’s general theory of relativity states that the curvature of spacetime dictates the way in which matter flows through it, and conversely, matter curves spacetime around it. Einstein’s theory explained the shift in Mercury’s perihelion, and so far seems to be the correct description of the motion of planets, stars and even galaxies.

**Gravitational waves: a new window into the Universe**

The general theory of relativity does more than just predicting the motion of objects. It also predicts a new type of radiation, known as gravitational radiation or gravitational waves. Gravitational waves are ripples in the curvature of spacetime, which propagate at the speed of light. The effect of gravitational waves is the periodic expansion and contraction of space and time (see Fig. 1).

**Figure 1: Example of the distortion of spacetime due to a incident gravitational wave onto a ring of test particles. The top and bottom row represent the effects of the two polarisation states as a function of the phase of the gravitational wave.**

The existence of gravitational waves has only been inferred indirectly through the motion of two stars orbiting each other. In particular, in 1974, Russell Hulse and Joseph Taylor found two pulsars (neutron stars that emit highly collimated beams of electromagnetic radiation) in a binary system that appeared to behave exactly as if the system was loosing energy and angular momentum in the form of gravitational waves (see Fig. 2). Today this discovery is regarded as the first indirect evidence of gravitational waves, and earned Hulse and Taylor the 1993 Nobel Prize in Physics [1].

**Figure 2: Change in the time of the periastron of the binary pulsar “PSR B1913+16” as a function of time (red dots). These observation are compared to the prediction of general relativity (blue line). This data is considered as the first indirect evidence of gravitational waves.**

**Quest for strong gravity**

So does this mean that general relativity has been fully verified? From a theoretical perspective, we might be inclined to say that general relativity cannot be the final answer, because of the current inability to describe it using a complete quantum theory. Therefore, it is currently not possible to unify gravity with the other forces of nature (electromagnetic, strong and weak force) into a grand unified theory, which some might argue is an indication that general relativity cannot be the final answer. In other words, a theory must exist that, in the low-energy regime, behaves like general relativity.

From an experimental perspective, one can argue that all of the tests of general relativity have so far been done in the regime of weak gravity. A figure of merit which describes the strength of gravity is the quantity ϵ ~ GM∕(Rc

^{2}), where G is the gravitational constant, M is the total mass of the system, R is the characteristic length scale of the system and c is the speed of light. Near a black hole the strength is ϵ ~ 10

^{-1}, whereas for solar system tests, and for binary pulsar tests, this strength is about ϵ ~ 10

^{-6}[2]. Therefore, there is a whole new regime of gravity to explore experimentally.

Scientists all over the world are working hard on the quest for strong gravity. Amongst many interesting questions, they also hope to uncover empirical insight into the quantisation of gravity, which could refine or guide new theories of gravity. One of the ways in which we could hope to probe the regime of strong gravity is through the direct measurement of gravitational waves. Such measurements could probe gravity close to black holes and other exotic astrophysical objects.

**Advanced LIGO and Virgo**

Large-scale physics experiments such as the USA-based LIGO (see Fig. 3) [3] and the Italy-based Virgo [4] aim to, for the first time in the history of mankind, detect the influences of gravitational waves directly. These experiments are set up to measure tiny changes in distances of about one thousandth of the diameter of a proton. These tiny perturbations of spacetime could lead us down a new path in our quest for strong gravity.

**Figure 3: Aerial view of the LIGO-Hanford detector**

The motion of the source closely dictates the characteristics of the gravitational waves emitted. So by mapping out the distortions caused by the incident gravitational wave, one could infer a wealth of information about its origins. In other words, where astronomers needed telescopes to determine the motion of planets, stars and galaxies, measurements of gravitational waves can provide an additional way to map the dynamics of celestial objects.

In particular, a promising class of candidates for the first detection of gravitational wave is the compact binary coalescence [5]. Compact binary coalescence typically refers to (especially in the context of LIGO/Virgo) the mergers of binary black holes or neutron stars (see Fig. 4). The components of such systems spiral toward each other as energy and momentum are radiated away through the emission of gravitational waves. Finally, when the objects are sufficiently close to each other, they merge to form a single black hole which then continues to ring down as it reaches a quiescence state. The dynamics of coalescence of a compact binary can be seen through simulations as in Ref. [6].

**Figure 4: Image from a binary black hole simulation.**

**Testing strong-field gravity with compact binary coalescences**

Compact binary coalescences are attractive systems to probe strong gravity, because close to black holes and neutron stars the effects of gravity can be considered strong. Moreover, these systems are relatively easy to understand theoretically, because they mainly involve the application of general relativity. In contrast, mechanisms behind, for example, supernovae, which are also candidates to be measured by Advanced LIGO/Virgo, involve a complicated interplay amongst many branches of physics. The direct measurement of gravitational waves emitted from a compact binary coalescence will therefore give us access to the motion of black holes in orbit around each other’s strong gravitational pull.

However, in order to extract this information, we need specialised algorithms to dig deep into the data. One of such algorithms is called Test Infrastructure for GEneral Relativity (TIGER). This algorithm tries to answer the question “is the signal consistent with general relativity?” through the application of Bayesian hypothesis testing [7]. This framework ensures the optimal use of available information, and allows one to combine information across multiple detections of compact binary coalescences. Using this algorithm in a simulation environment, we have shown that the Advanced LIGO-Virgo network is indeed capable of probing gravity in uncharted territories, to an accuracy never seen before.

Of course, many challenges have to be faced. Detection of gravitational waves is a major challenge by itself on which hundreds of scientist are currently working. Moreover, once the Advanced LIGO-Virgo network is making confident detections, we need to analyse the motion of black holes or neutron stars in the presence of noise that can be orders of magnitude louder than the signal. Nevertheless, we are on the brink of the first direct detection with Advanced LIGO coming online as early as 2015. A hundred years after the introduction of general relativity, Advanced LIGO/Virgo could either put the crown on Einstein’s work, or showcase its limitations.

To be continued…

**References**

**[1]**“The Nobel Prize in Physics 1993”. Link in: Nobelprize.org.

**[2]**C. M. Will. “The Confrontation between General Relativity and Experiment”. In: Living Reviews in Relativity 17.4 (2014). Link.

**[3]**http://www.ligo.org .

**[4]**http://wwwcascina.virgo.infn.it .

**[5]**B.S. Sathyaprakash and Bernard F. Schutz. “Physics, Astrophysics and Cosmology with Gravitational Waves”. In: Living Reviews in Relativity 12.2 (2009). Link.

**[6]**Download from: http://numarch.aei.mpg.de/numrel-webpages/movies/bbh08_small.mov .

**[7]**T. G. F. Li, W. Del Pozzo, S. Vitale, C. Van Den Broeck, M. Agathos, J. Veitch, K. Grover, T. Sidery, R. Sturani, A. Vecchio, “Towards a generic test of the strong field dynamics of general relativity using compact binary coalescence”. Physical Review D, 85, 082003 (2012). Abstract.

Labels: Gravitation 3, Gravitational Waves 3

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