General Relativity Problems with special relativity What makes inertial frames special? How do you determine whether a frame is inertial? Inertial to what? Problems with gravity: In equation F = GM 1M 2, the value r depends r on the frame of reference! 2 Instantaneous change in force if object s move? How does other mass find out so quickly? Why is inertial mass equal to gravitational mass? Coincidence?
The Equivalence Principle Consider a person in a spaceship at rest out in space, far from any source of gravity. Compare what that person feels to a person free-falling to the Earth s surface.
Equivalence Principle Now consider a person stationary on the surface of the Earth, and compare to a person in an accelerating rocket (far from any source of gravity).
Equivalence Principle Einstein s Equivalence Principle: It is impossible to tell by experiment whether you are in accelerated frame of reference or in a gravitational field. The effects of gravity is completely indistinguishable from effects of an accelerating frame
Equivalence Principle Consider the trajectory of a thrown ball in both frames of reference (left frame is accelerating, right frame is stationary near Earth)
Equivalence Principle Conversely, imagine a ball thrown sideways in both a free-falling frame (near Earth) and an elevator at rest far from any source of gravity:
Equivalence Principle Why do objects with different masses fall at the same rate? Easy to see why in an accelerating elevator: Floor gains on all objects at the same rate since it is the floor that is accelerating upward (the objects are not accelerating). a
Consequences of Equivalence Principle Mass must bend light! (Reason photons bends in accelerating elevator is identical to reason a thrown ball bends) On Earth the effect is incredibly small (too small to measure, like 10-13 degrees in a 10 m laboratory). y = 1 2 gt2 = 1 2 g(l/c)2 tan θ = y L = 1 gl 2 c 2 3 10 16 Rad (on Earth) On neutron star, if L=5 m, θ 1.1 10 4 Rad 0.4
Gravitational Lens 1915: Einstein predicted observable effect due to Sun: Proper treatment requires integration over lightpath, accounting for changing values of g. Angle of deflection θ = 4GM Rc 2 =1.7 Confirmed during solar eclipse by Eddington (1919)
Gravitational Redshift Consider accelerating elevator: Detector s velocity when receiving signal is different than source s velocity when emitting signal. v a t ah/c Doppler shift causes observed wavelength to be longer than emitted wavelength Detector a v 2 Detector v 1 a λ λ v c ah c 2 v 1 source v 2 source
Gravitational Redshift Same thing should occur on Earth (due to the equivalence principle). Detector a distance h above Earth s surface should see a gravitational redshift: Detector λ λ gh c 2 source
Gravitational Time Dilation The phenomena of gravitational redshift is attributed to time dilation Which clock ticks more quickly? a) Clock #1 b) Clock #2 c) They tick at the same rate! Clock #1 Clock #2
Geometry The basic idea of general relativity is that matter effect s the geometry of space-time. Matter does not produce a force on an object. Matter causes space-time to be curved (or warped) such that straight trajectories (resulting from a coasting object) appear curved when projected on a flat canvas. What do we mean by straight? For 3D geometry in a Euclidean ( flat ) coordinate system, a straight line is the trajectory in which the distance between two points is a minimum l = x 2 + y 2 + z 2
Flat Space-Time (SR) For GR, we have to consider the curvature of 4-dimensional space-time. Minkowski space-time metric is flat (similar to Euclidean 3d) s 2 =(c t) 2 ( x 2 + y 2 + z 2 ) s 2 =(c t) 2 ( r 2 + r 2 θ 2 + r 2 sin 2 θ φ 2 ) s = c2 1 ds = dt 2 dl 2 = (v/c)2 cdt ct (x 1,ct 1 ) (x 2,ct 2 ) x Δs is frame invarient: Proper time. In frame where v=0, An object coasting has a trajectory in which the overall space-time interval is a minimum. This gives s = c τ x(τ),y(τ),z(τ), and t(τ)
Curved Space-Time Mass warps space and time. The metric must contain curvature terms: 3 3 s 2 = g ij x i x j = g ij x i x j i=0 j=0 s 2 = g 00 t 2 + g 11 x 2 + g 22 y 2 + g 33 z 2 + g 12 x y + g 13 x z + g 23 y z + g 01 t x For flat spacetime, g 00 = c 2, g 11 = g 22 = g 33 = 1 Values (or functions) of the gs different than these values represent curved space-time An object coasting has a trajectory in which the overall spacetime interval is a minimum. This gives (after application of calculus of variations) x(τ),y(τ),z(τ), and t(τ)
Curved Surface What is the shortest trajectory to fly from Denver to London? If you look at trajectory on a Euclidean map (ignoring curvature of Earth), trajectory is curved!
Schwarzschild Metric Schwarzschild found a solution to Einstein s field equations for outside a spherical object of mass M and radius R. s 2 = R rc 2 c 2 t 2 r2 rc 2 r 2 ( θ 2 +sin 2 θ φ 2 ) r r,θ,φ, and t are coordinates a far-away observer (ignorant of curvature) uses. All observers agree on the value of the spacetime interval, s
Gravitational Time Dilation Revisited s 2 = rc 2 Consider a stationary person a distance r from center of spherical object. This person measures his proper time, Δτ =Δs/c. If the observer doesn t move, then Thus the reading of the far-away clock, Δt, differs from the reading of the person s clock Δτ via τ = Gravitational redshift: c 2 t 2 r2 rc 2 r 2 ( θ 2 +sin 2 θ φ 2 ) rc 2 s = c τ = t ν 1 t rc 2 c t ν = ν 0 rc 2 λ 1 = λ 0 rc 2
Gravitational Length Expansion Now consider two points located at r 1 and r 2 (in line with the center of the spherical mass). What is the distance between the two points? Far-away ignorant observer would say L=Δr (=r 2 -r 1 ). But Schwarzschild metric now gives: r s = rc 2 Actual space is stretched due to mass. = L
Time Delay of light travel Consider motion of photon moving from position r 1 to r 2 M R r 1 r 2 Spacetime interval for photon is zero: ds 2 =0=c 2 dt 2 c 2 dr2 r rc 2 t = t = r 2 r 1 c dt = 1 c r2 r 1 + 2GM c 3 dr rc 2 r2 2GM c ln 2 r c 2 light takes longer (according to our clock) due to both space stretching and time dilation
Trajectories Near Compact Object Consider purely radial motion ds 2 = rc 2 c 2 dt 2 dr2 rc 2 s = rc 2 c 2 dt 2 dr2 rc 2 = rc 2 c 2 ( t/ τ) 2 1 rc 2 ( r/ τ) 2 dτ Using calculus of variations to find an extremum, the functions r(τ) and t(τ) can be determined [and thus ] r(t) Mike Guidry, Univ. Tenn,Knoxville
Experimental Tests of Time Dilation 1959 Pound-Rebka Esperiment: Gravitational redshift measured at Harvard (a 22 m building) using Iron-57 gamma rays. Change in frequency is ν ν 2 10 15 Echo delay off Venus (Shapiro Delay); pulse is 0.1 ms delayed in the 20 minute round trip) (1966) GPS devices require the use of GR for sub 1-meter positioning accuracy. 2 10 9 ν ν Synchronized clocks, with one placed on airplane or satellite, don t remain synchronized. Gravitational redshift detected on neutron stars and white dwarfs.