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Home , Karl Schwarzschild

The Basics

Credit: SXS Lensing; http://www.black-holes.org/

The Laser Interferometer Gravitational-wave Observatory: a Caltech/MIT collaboration supported by the National Science Foundation Gregory Mendell, LIGO Hanford Observatory

LIGO-G1801389 Einstein Wondered:

Can we catch light?

Photo: at the first Solvay Conference, 1911; Public Domain Niels Bohr Albert Einstein

Time Dilation

Δx=vΔt T Δ c ΔT= Δt 1-v2/c2 Δ = change in T = time measured by motorcycle riders t = time measured by observer at “rest” v = speed of motercycles c = LIGO-G0900422-v1Warning: thought experiment only; do not try this at home. Start Motorcycle: http://en.wikipedia.org/wiki/Motorcycle_racing The Pythagorean Theorem Of c2ΔT2 + v2Δt2 = c2Δt2 c2ΔT2 = c2Δt2 - v2Δt2 c2ΔT2 = c2Δt2 - Δx2 c = 1 light-year/year

ΔT2 = Δt2 - Δx2

(Usually known as the spacetime interval)

Example

t Spacetime c = 1 light-year/year

ΔT2 = Δt2 - Δx2

Δt = 30 years; Δx = 29 lt-yrs. v = 96.7% the speed of light

2 2 2 2 30= years t

ΔT = 30 – 29 = 59 yrs Δ ΔT = 7.7 years Δx = 29 light-years x

Einstein’s Happiest Thought

Gravity

J J disappears when you free fall!

Photo: NASA http://en.wikipedia.org/wiki/ Leaning_Tower_of_Pisa

LIGO-G0900422-v1 Warning: thought experiment only; do not try this at home. Einstein’s General

8πG Δ àd = infinitesimal G = T µν c4 µν change 1 G = R − g R µν µν 2 µν 2 2 2 dT = g dt + g dx α µν tt xx Rµν = R µαν ; R = g Rµν α α α α γ α γ 2 µ ν R µβν = ∂ β Γ µν − ∂ν Γ µβ + Γ βγ Γ µν − Γ γν Γ µβ dT = gµνdx dx α 1 αβ Γ µν = g (∂ν gµβ + ∂ µ gβν − ∂ β gµν ) In GR the components of 2 a 4x4 symmetric matrix Einstein’s Field Equations called the metric tensor dxα define the curvature of =U α ; U = g U β U = 4-Vel.; T = dT α αβ spacetime. dU 1 α = ∂ g U βU γ Geodesic Equation dT 2 α βγ Horizon Black Holes are: Our Universe

2. Holes in Space Time Credit: Alain Riazuelo, IAP/UPMC/CNRS Space

1. Black 0 Escape Velocity = Speed of Light 3. Space & Time Warps Schwarzschild 1916; Einstein & Rosen 1935; many others in the 1960s Schwarzschild

2 2 ⎛ 2GM ⎞ 2 2 1 2 2 2 2 2 2 c dT = ⎜1− 2 ⎟c dt − dr − r dθ − r sin θdφ ⎝ rc ⎠ ⎛ 2GM ⎞ ⎜1− 2 ⎟ ⎝ rc ⎠ ⎛ v2 ⎞ 1 c2dT 2 = ⎜1− esc ⎟c2dt 2 − dr 2 − r 2dθ 2 − r 2 sin2 θdφ 2 ⎜ c2 ⎟ ⎛ v2 ⎞ ⎝ ⎠ ⎜1− esc ⎟ ⎜ 2 ⎟ Karl Schwarzschild ⎝ c ⎠ Object 2GM • Escape vesc = You 1 thousand, million, r Velocity million, millionth the 2GM • Schwarzschild thickness of a human hair R = s c2 Radius Earth 1 cm (size of marble) Sun 3 km (2 miles)‏

LIGO-G0900422-v1 Gravitational

2GM ΔT = 1− Δt rc2 slows time down!

Photo:http://en.wikipedia.org/wiki/ Clock_Photos:http://en.wikipedia.org/ Leaning_Tower_of_Pisa LIGO-G0900422-v1 wiki/Cuckoo_clock Black Hole Embedding Diagram Schwarzschild solution to General Relativity for t = constant, θ = π /2: 1 ds2 = dr2 + r2dϕ 2 " 2GM % $1− ' # rc2 & z The Schwarzschild or Einstein-Rosen Bridge (Flamm 1916, Physikalische Zeitschrift. XVII: 448; Einstein & Rosen 1935, Phys. Rev. 48 73; y Misner & Wheeler 1957, Ann. x Phys. 2: 525)

LIGO-G0900422-v1 Non-traversable Wormhole

Our Universe

Another Universe LIGO-G0900422-v1 Gravitational Waves

Gravitational waves are ripples in the fabric of spacetime, stirred up by the changing motions of matter and energy.

Binary Black Hole Merger Simulation

Credit: SXS Collaboration/Canadian Institute for Theoretical Astrophysics/SciNet; http://www.black-holes.org/

LIGO-G1600258 Gravitational Waves The LIGO Detectors

The waves are extremely weak by the time they reach Earth: ΔL/L ~ 10-21 The End LASER

PHOTODIODE Detector Response µ υ gµυdx dx = 0 (Light Travels On Null Geodesics)‏

⎛1+ hxx hxy hxz ⎞⎛dx ⎞ 2 2 ⎜ ⎟⎜ ⎟ c dt − (dx 0 0)⎜ hyx 1+ hyy hyz ⎟⎜ 0 ⎟ = 0 ⎜ h h 1 h ⎟⎜ 0 ⎟ ⎝ zx zy + zz ⎠⎝ ⎠ 2 2 2 c dt = (1+ hxx )dx Δt L L ⎛ 1 ⎞ c dt = 1+ hxx dx ≅ ⎜1+ hxx ⎟dx ∫0 ∫0 ∫0 ⎝ 2 ⎠ L cΔt = L = L + h x 2 xx ΔL 1 = (hxx −h yy ) = F+ (θ,φ,ψ )h+ (t) + F× (θ,φ,ψ )h× (t) L 2 Summary

• , 1905: nothing can faster than light and the faster you go the slower time goes. Space and time become spacetime! • Einstein’s happiest thought, 1907: gravity disappears when you free fall. • General Relativity, 1915: gravity is a warping of space and time. • The Schwarzschild Solution, 1916: Black Holes! • Gravitational Waves, 1916: the changing motion of matter and energy can produce ripples in the fabric of spacetime.

Photo: Albert Einstein at the first Solvay Conference, 1911; Public Domain Would you believe you could fall into hole in completely empty space, a hole from which nothing can escape, not even light? Gravitational Waves

dT 2 = g dx µdxν Gravitational waves carry µν g =η + h information about the µν µν µν ⎛ 1 ∂2 ⎞ 2 h µν 0 spacetime around black ⎜∇ − 2 2 ⎟ = ⎝ c ∂t ⎠ 1 holes & other sources. hTT (θ = π / 2) ∝ cos[2πf (t − r / c) + 2φ] θˆθˆ r

⎛0 0 0 0⎞ ⎜ ⎟ 0 h h 0 hTT ⎜ + × ⎟e2πif (t−z / c) µν = ⎜ ⎟ 0 h× − h+ 0 ⎜ ⎟ ⎝0 0 0 0⎠

LIGO-G0900422-v1 Gravitational Waves

Gravitational waves are ripples in the fabric of spacetime, stirred up by the changing motions of matter and energy. The waves are extremely weak by the times they reach Earth.