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Gravitational Waves

Field: General Relativity

Ripples in spacetime caused by some of the most violent and energetic processes in the Universe.

Sequence of Expressions

The intrinsic curvature of a spacetime manifold (M,g)(M, g) is quantified by the Riemann curvature tensor RμνρσR_{\mu\nu\rho\sigma}. This tensor is defined in terms of the Christoffel symbols Γμνλ\Gamma^\lambda_{\mu\nu} (which are derived from the metric gμνg_{\mu\nu}) by:\n\nRμνρσ=12(∂ρ∂νgμσ−∂σ∂νgμρ−∂ρ∂μgνσ+∂σ∂μgνρ)R_{\mu\nu\rho\sigma} = \frac{1}{2} (\partial_{\rho} \partial_{\nu} g_{\mu\sigma} - \partial_{\sigma} \partial_{\nu} g_{\mu\rho} - \partial_{\rho} \partial_{\mu} g_{\nu\sigma} + \partial_{\sigma} \partial_{\mu} g_{\nu\rho})\n\nAlternatively, using the Christoffel symbols: Rμνρσ=gσλ(∂ρΓνλ−∂νΓρλ+ΓρλΓνσ−ΓνλΓρσ)R_{\mu\nu\rho\sigma} = g_{\sigma\lambda} (\partial_{\rho} \Gamma^\lambda_{\nu} - \partial_{\nu} \Gamma^\lambda_{\rho} + \Gamma^\lambda_{\rho} \Gamma^\sigma_{\nu} - \Gamma^\lambda_{\nu} \Gamma^\sigma_{\rho}).
Define the Einstein tensor GμνG_{\mu\nu} as Gμν=Rμν−12gμνRG_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R, where RμνR_{\mu\nu} is the Ricci curvature tensor and RR is the Ricci scalar. The field equations relate this geometric tensor to the energy-momentum tensor TμνT_{\mu\nu} via:\n\nGμν=8πGc4TμνG_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}\n\nHere, gμνg_{\mu\nu} is the metric tensor, GG is Newton's gravitational constant, and cc is the speed of light.
For a static, spherically symmetric mass MM in vacuum, the Schwarzschild metric gμνg_{\mu\nu} in coordinates (t,r,θ,ϕ)(t, r, \theta, \phi) is given by:\n\nds2=gμνdxμdxν=−(1−2GMc2r)c2dt2+1(1−2GMc2r)dr2+r2(dθ2+sin⁡2θdϕ2)ds^2 = g_{\mu\nu} dx^\mu dx^\nu = -\left(1 - \frac{2GM}{c^2 r}\right) c^2 dt^2 + \frac{1}{\left(1 - \frac{2GM}{c^2 r}\right)} dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\phi^2)\n\nWhere GG is the gravitational constant, MM is the mass, and cc is the speed of light. The components are gtt=−(1−2GMc2r)g_{tt} = -\left(1 - \frac{2GM}{c^2 r}\right), grr=(1−2GMc2r)−1g_{rr} = \left(1 - \frac{2GM}{c^2 r}\right)^{-1}, and gθθ=r2g_{\theta\theta} = r^2.
Let xμ(λ)x^\mu(\lambda) be the path of a freely falling particle parameterized by λ\lambda. The geodesic equation describes this path by minimizing the proper time interval and is given by:\n\nd2xμdλ2+Γνρμdxνdλdxρdλ=0\frac{d^2 x^\mu}{d\lambda^2} + \Gamma^\mu_{\nu\rho} \frac{d x^\nu}{d\lambda} \frac{d x^\rho}{d\lambda} = 0\n\nWhere Γνρμ\Gamma^\mu_{\nu\rho} are the Christoffel symbols of the second kind, defined as Γνρμ=12gμσ(∂νgρσ+∂ρgνσ−∂σgνρ)\Gamma^\mu_{\nu\rho} = \frac{1}{2} g^{\mu\sigma} (\partial_{\nu} g_{\rho\sigma} + \partial_{\rho} g_{\nu\sigma} - \partial_{\sigma} g_{\nu\rho}).
Define the Riemann curvature tensor RR for a manifold with metric gμνg_{\mu\nu} as:\nRiem(X,Y)Z=∇X∇YZ−∇Y∇XZ−∇[X,Y]Z\text{Riem}(X, Y) Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z \nIn local coordinates, the components are given by:\nRρσμν=∂Γνρ∂xμ−∂Γμρ∂xν+ΓνλΓλρ−ΓμλΓλρR^{\rho}{}_{\sigma\mu\nu} = \frac{\partial \Gamma^{\rho}_{\nu}}{\partial x^{\mu}} - \frac{\partial \Gamma^{\rho}_{\mu}}{\partial x^{\nu}} + \Gamma^{\lambda}_{\nu} \Gamma^{\rho}_{\lambda} - \Gamma^{\lambda}_{\mu} \Gamma^{\rho}_{\lambda}
For two inertial frames SS and S′S' moving relative to each other with a constant relative velocity v\mathbf{v}, the transformation of coordinates xμ=(ct,x,y,z)x^{\mu} = (ct, x, y, z) is given by the Lorentz transformation matrix Λμν\Lambda^{\mu}{}_{\nu}:\nx′μ=Λμνxνx'^{\mu} = \Lambda^{\mu}{}_{\nu} x^{\nu} \nFor a boost along the xx-axis, the matrix elements are:\nΛ=(γ−γβ00−γβγ0000100001)\Lambda = \begin{pmatrix} \gamma & -\gamma \beta & 0 & 0 \\ -\gamma \beta & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \nWhere β=v/c\beta = v/c and γ=11−β2\gamma = \frac{1}{\sqrt{1 - \beta^2}}.
Consider two observers, A (staying in an inertial frame) and B (accelerating and returning to A's frame). The proper time τ\tau elapsed for B, relative to the coordinate time tt measured by A, is governed by the time dilation factor γ(v)\gamma(v): \nΔτB=∫tstarttend1−v(t)2c2 dt\Delta \tau_B = \int_{t_{start}}^{t_{end}} \sqrt{1 - \frac{v(t)^2}{c^2}} \ dt \nThe paradox is resolved by recognizing that the integral for ΔτB\Delta \tau_B must be calculated using the non-inertial path (acceleration) taken by B, leading to ΔτB<ΔτA\Delta \tau_B < \Delta \tau_A.
For a stationary clock at a radial distance rr from a mass MM in a weak gravitational field, the relationship between the proper time dτd\tau measured by the clock and the coordinate time dtdt measured by an observer far away (r→∞r \to \infty) is derived from the metric component g00g_{00}:\ndτdt=−g00≈1−2GMrc2\frac{d\tau}{dt} = \sqrt{-g_{00}} \approx \sqrt{1 - \frac{2GM}{rc^2}} \nThis shows that the rate of time passage decreases as the gravitational potential increases (i.e., as rr decreases).
Consider a test mass mm undergoing free fall in a gravitational field. The Principle of Equivalence states that the trajectory of this mass is independent of its internal composition and structure. Mathematically, this implies that the equation of motion derived from the gravitational force F⃗g\vec{F}_g is identical to the equation of motion derived from an inertial force F⃗i\vec{F}_i in an accelerating reference frame:\n\nd2xdt2=g⃗andd2xdt2=a⃗inertial\frac{d^2 x}{d t^2} = \vec{g} \quad \text{and} \quad \frac{d^2 x}{d t^2} = \vec{a}_{\text{inertial}} \n\nThus, the gravitational acceleration g⃗\vec{g} is locally indistinguishable from the acceleration a⃗\vec{a} of the reference frame.
For a system described by the stress-energy tensor TμνT_{\mu\nu}, the local conservation of energy and momentum is expressed by the covariant divergence equation:\n∇μTμν=0\nabla_{\mu} T^{\mu\nu} = 0 \nWhere ∇μ\nabla_{\mu} is the covariant derivative associated with the metric gμνg_{\mu\nu}. This equation ensures that the total energy-momentum remains constant in spacetime.