Einstein Field Equations

Einstein derived his equations in November 1915, one week before Hilbert. A century later LIGO (2015) heard gravitational waves from the merger of two black holes of 36 and 29 solar masses -- the signal lasted 0.2 seconds and traveled 1.3 billion years.

• Cosmology: FLRW + Friedmann equation models cosmic expansion
• GPS: corrections for gravitational time dilation (Schwarzschild) -- 45 microseconds/day
• Gravitational wave observatories: LIGO, Virgo, LISA detect mergers

Предварительные знания

• Previous lesson

Einstein Field Equations

Einstein in 1915 derived the equations G_mu_nu + Lambda g_mu_nu = 8*pi*G T_mu_nu, where G_mu_nu = R_mu_nu - (1/2)*R*g_mu_nu is the Einstein tensor and T_mu_nu is the stress-energy tensor. Ten nonlinear PDEs for ten metric components g_mu_nu. The first exact solution was Schwarzschild (1916), describing geometry around mass M.

Linearized Gravity and Gravitational Waves

Linearization: g_mu_nu = eta_mu_nu + h_mu_nu, |h| << 1. In Lorenz gauge: box h_bar_mu_nu = -16*pi*G T_mu_nu, where h_bar_mu_nu = h_mu_nu - (1/2) eta_mu_nu h. Solution: gravitational waves propagating at speed c. LIGO (2015) detected black hole merger: strain delta L/L ~ 10^{-21}.

Key Ideas

• G_mu_nu + Lambda g_mu_nu = 8pi G T_mu_nu -- 10 nonlinear PDEs for g_mu_nu
• Schwarzschild: r_s = 2GM/c^2, g_tt = -(1-r_s/r), g_tt(r_s) = 0
• Friedmann: H^2 = (8pi G/3) rho - k/a^2 + Lambda/3
• Linearization: box h_bar_mu_nu = -16pi G T_mu_nu in Lorenz gauge
• Quadrupole formula: P = (G/(5c^5)) <Q_ij triple-dot Q^ij triple-dot>

Further Directions

These ideas open paths to deeper mathematics.

• de-28-wave-equation — extends

Вопросы для размышления

• Give a concrete example.
• How does this connect to other areas of mathematics?