Part Two by Lydia Bieri, David Garfinkle, and Nicolás Yunes Gravitational Waves Are Vibrations in Spacetime Propagating At

Part two by Lydia Bieri, David Garfinkle, and Nicolás Yunes Gravitational Waves and Their Mathematics Introduction In 2015 gravitational waves were detected for the first time by the LIGO team [1]. This triumph happened 100 years after Albert Einstein’s formulation of the theory of general relativity and 99 years after his prediction of gravitational waves [4]. This article focuses on the mathematics of Einstein’s gravitational waves, from the properties of the Einstein vacuum equations and the initial value problem (Cauchy problem), to the various approximations used to obtain quantitative predictions from these equations, and eventually an experimental detection. General relativity is studied as a branch of astronomy, physics, and mathematics. At its core are the Einstein equations, which link the physical content of our universe to geometry. By solving these equations, we construct the spacetime itself, a continuum that relates space, time, geometry, and matter (including energy). The dynamics of the gravitational field are studied in the Cauchy problem for the Einstein equations, relying on the theory of nonlinear partial differential equations (pde) and geometric analysis. The connections between astronomy, physics, and mathematics are richly illustrated by the story of gravitational radiation. In general relativity, the universe is Figure 1. Albert Einstein predicted gravitational described as a space- Gravitational waves in 1916. time manifold with a curved metric whose waves are curvature encodes the properties of the grav- vibrations in to study and nothing else. We might consider the solar itational field. While system as an isolated object, or a pair of black holes sometimes one wants spacetime spiraling into one another until they collide. We ask how to use general relativity to describe the propagating at the those objects look to a distant, far away observer in a whole universe, often speed of light. region where presumably the curvature of spacetime is we just want to know very small. Gravitational waves are vibrations in space- how a single object or time that propagate at the speed of light away from their small collection of objects behaves. To address that kind source. They may be produced, for example, when black of problem, we use the idealization of the isolated sys- holes merge. This is what was detected by Advanced LIGO tem: a spacetime consisting of just the objects we want (aLIGO) and this is the focus of this article. First we describe the basic differential geometry used Lydia Bieri is associate professor in the Department of Mathemat- to define the universe as a geometric object. Next we ics at the University of Michigan in Ann Arbor. Her e-mail address describe the mathematical properties of the Einstein vac- is [email protected]. uum equations, including a discussion of the Cauchy David Garfinkle is professor in the Physics Department of Oakland University in Michigan, and also a visiting research problem and gravitational radiation. Then we turn to the scientist at the University of Michigan. His e-mail address is various approximation schemes used to obtain quantita- [email protected]. tive predictions from these equations. We conclude with Nicolás Yunes is associate professor of physics at Montana State the experimental detection of gravitational waves and the University. His e-mail address is [email protected]. astrophysical implications of this detection. This detec- For permission to reprint this article, please contact: tion is not only a spectacular confirmation of Einstein’s [email protected]. theory, but also the beginning of the era of gravitational DOI: http://dx.doi.org/10.1090/noti1549 wave astronomy, the use of gravitational waves to investi- August 2017 Notices of the AMS 693 gate aspects of our universe that have been inaccessible and spacelike if 2 to telescopes. 푔푥(푋, 푋) > 0. In general relativity nothing travels faster than the speed The Universe as a Geometric Object of light, so the velocities of massless particles are null A spacetime manifold is defined to be a 4-dimensional, vectors whereas those for massive objects are timelike. oriented, differentiable manifold 푀 with a Lorentzian A causal curve is a differentiable curve for which the metric tensor, 푔, which is a nondegenerate quadratic tangent vector at each point is either timelike or null. form of index one, 3 휇 휈 푔 = ∑ 푔휇휈푑푥 ⊗ 푑푥 , 휇,휈=0 defined in 푇푞푀 for every 푞 in 푀 varying smoothly in 푞. The trivial example, the Minkowski spacetime as defined in Einstein’s special relativity, is ℝ4 endowed with the flat Minkowski metric: (1) 푔 = 휂 = −푐2푑푡2 + 푑푥2 + 푑푦2 + 푑푧2. Taking 푥0 = 푡, 푥1 = 푥, 푥2 = 푦, and 푥3 = 푧, we have 2 휂00 = −푐 , 휂푖푖 = 1 for 푖 = 1, 2, 3, and 휂휇휈 = 0 for 휇 ≠ 휈. In mathematical general relativity we often normalize the speed of light, 푐 = 1. Figure 2. Light cones, a timelike curve, and a The family of spacelike hypersurface as demonstrated by physics Schwarzschild metrics majors at Lehman College. Schwarzschild are solutions of the Einstein vacuum equa- A hypersurface is called spacelike if its normal vector tions that describe spacetime is timelike, so that the metric tensor restricted to the spacetimes containing hypersurface is positive definite. A Cauchy hypersurface describes a black a black hole, where is a spacelike hypersurface where each causal curve the parameter values through any point 푥 ∈ 푀 intersects ℋ exactly at one hole. are 푀 > 0. Taking point. A spacetime (푀, 푔) is said to be globally hyperbolic 푟 = 2퐺푀/푐2, it has 푠 if it has a Cauchy hypersurface. In a globally hyperbolic the metric: spacetime, there is a time function 푡 whose gradient is 2 푟푠 everywhere timelike or null and whose level surfaces (1 − 4휌 ) 4 2 2 푟푠 (2) 푔 = −푐 2 푑푡 + (1 + 4휌 ) ℎ, are Cauchy surfaces. A globally hyperbolic spacetime is 푟푠 (1 + 4휌 ) causal in the sense that no object may travel to its own past. where 휌2 = 푥2 + 푦2 + 푧2, ℎ = 푑푥2 + 푑푦2 + 푑푧2, and 퐺 As in Riemannian geometry, curves with 0 acceleration denotes the Newtonian gravitational constant. This space are called geodesics. Light travels along null geodesics. is asymptotically flat as 휌 → ∞. Geodesics that enter the event horizon of a black hole as The Friedmann-Lemaître-Robertson-Walker spacetimes in Figure 3 never leave. Objects in free fall travel along describe homogeneous and isotropic universes through timelike geodesics. They also can never leave once they the metric have entered a black hole. When two black holes fall into 2 2 2 (3) 푔 = −푐 푑푡 + 푎 (푡)푔휒, each other, they merge and form a single larger black hole. where 푔휒 is a Riemannian metric with constant sectional curvature, 휒, (e.g. a sphere when 휒 = 1) and 푎(푡) describes In curved spacetime, geodesics bend together or apart the expansion of the universe. The function, 푎(푡), is found and the relative acceleration between geodesics is de- by solving the Einstein equations as sourced by fluid scribed by the Jacobi equation, also known as the matter. geodesic deviation equation. In particular, the relative In an arbitrary Lorentzian manifold, 푀, a vector 푋 ∈ acceleration of nearby geodesics is given by the Riemann curvature tensor times the distance between them. The 푇푥푀 is called null or lightlike if Ricci curvature tensor, 푅휇휈, measures the average way 푔푥(푋, 푋) = 0. in which geodesics curve together or apart. The scalar At every point there is a cone of null vectors called the null curvature, 푅, is the trace of the Ricci curvature. cone, as in Figure 2. A vector 푋 ∈ 푇푥푀 is called timelike if Einstein’s field equations are: 8휋퐺 푔푥(푋, 푋) < 0, (4) 푅 − 1 푅푔 = 푇 , 휇휈 2 휇휈 푐4 휇휈 2 Editor’s note: Don’t miss the intriguing and most readable final where 푇휇휈 denotes the energy-momentum tensor, which sections of this article. encodes the energy density of matter. Note that for 694 Notices of the AMS Volume 64, Number 7 Figure 3. The horizon of the black hole is depicted here as a cylinder with inward pointing light cones, as demonstrated by physics majors at Lehman College. cosmological considerations, one can add Λ푔휇휈 on the left-hand side, where Λ is the cosmological constant. However, nowadays, this term is commonly absorbed into 푇휇휈 on the right-hand side. Here we will consider the noncosmological setting. One then solves the Einstein equations for the metric tensor 푔휇휈. If there are no other fields, then 푇휇휈 = 0 and (4) reduce to the Einstein vacuum equations: (5) 푅휇휈 = 0 . Note that the Einstein equation is a set of second order quasilinear partial differential equations for the metric tensor. In fact, when choosing the right coordinate chart (wave coordinates), taking 푐 = 1, and writing out the formula for the curvature tensor, 푅 , in those 휇휈 Figure 4. Albert Einstein coordinates, the equation becomes: (6) □푔푔훼훽 = 푁훼훽 □ where 푔 is the wave operator and 푁훼훽 = 푁훼훽(푔, 휕푔) differential equation derived from (4). If a solution has a denote nonlinear terms with quadratics in 휕푔. time where the scale factor vanishes, then the solution Quite a few exact solutions to the Einstein vacuum is said to describe a cosmos whose early phase is a “big equations are known. Among the most popular are bang.” the trivial solution (Minkowski spacetime) as in (1); the Schwarzschild solution, which describes a static The Einstein Equations black hole, as in (2); and the Kerr solution, which de- Beginnings of Cauchy Problem scribes a black hole with spin angular momentum. Note that the exterior gravitational field of any spherically In order to study gravitational waves, stability problems, symmetric object takes the form of (2) for 푟 > 푟0 where and general questions about the dynamics of the gravita- 푟0 > 푟푠 is the radius of the object, so this model can be tional field, we have to formulate and solve the Cauchy used to study the spacetime around an isolated star or problem.

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