General Relativity Introduction GR is Einsteins theory of

General Relativity

Introduction GR is Einstein’s theory of gravitation that builds on the geometric concept of spacetime introduced in SR. Is there a more fundamental explanation of gravity than Newton’s law? GR makes specific predictions of deviations from Newtonian gravity.

Curved space-time Gravitational fields alter the rules of geometry in space-time producing “curved” space For example the geometry of a simple triangle on the surface of sphere is different than on a flat plane (Euclidean) On small regions of a sphere, the geometry is close to Euclidean

How does gravity curve space-time? • With no gravity, a ball thrown upward continues upward and the worldline is a straight line. • With gravity, the ball’s worldline is curved. No gravity t t x x • It follows this path because the spacetime surface on which it must stay is curved. • To fully represent the trajectory, need all 4 space-time dimensions curving into a 5 th dimension(!) • Hard to visualize, but still possible to measure

Principle of Equivalence A uniform gravitational field in some direction is indistinguishable from a uniform acceleration in the opposite direction Keep in mind that an accelerating frame introduces pseudo-forces in the direction opposite to the true acceleration of the frame (e. g. inside a car when brakes are applied)

Elevator experiment • First: elevator is supported and not moving, but gravity is present. Equate forces on the person to ma (=0 since a=0) • Fs - mg = 0 so Fs = mg • Fs gives the weight of the person. Let upward forces be positive, thus gravity is -g • Second: no gravity, but an upward acceleration a. The only force on the person is Fs and so • Fs = ma or Fs = mg if value of a is the same as g • Person in elevator cannot tell the difference between gravitational field and accelerating frame • Third: there is gravity and the elevator is also in free-fall • Fs - mg = -mg or Fs = 0 See also http: //www. pbs. org/wgbh/nova/einstein/relativity/ • “Weightless”

Dichotomy in the concept of "mass" “mass” a measure of an object’s resistance to changes in movement (F=ma) inertial mass “mass” a measure of an objects response to gravitational attraction (F=GMm/r 2) gravitational mass. Dichotomy resolved by putting gravity and acceleration on an equal footing. The principle of equivalence is really a statement that inertial and gravitational masses are the same for any object. This equivalence means that all objects have the same acceleration in a gravitational field (e. g. a feather and bowling ball fall with the same acceleration in the absence of air friction).

The GR equations relate the curvature of spacetime with the energy and momentum within the spacetime (Matter tells spacetime how to curve, and curved space tells matter how to move). Where and vary from 0 to 3, thus this equation really represents 16 equations Ricci curvature tensor - R Metric coefficients - g - relates length interval to coordinate system (matrix) Christoffel symbols – cross terms Gμν = 8πTμν = Rμν – 1/2 gμνR how space is curved location and motion of matter (energy-momentum tensor)

Tests of General Relativity Orbiting bodies - GR predicts slightly different paths than Newtonian gravitation Most obvious in elliptical orbits where distance to central body is changing and orbiting object is passing through regions of different space-time curvature The effect - orbit does not close and each perihelion has moved slightly from the previous position

In our Solar System, the effect is greatest for Mercury - closest to Sun and high eccentricity • Mercury’s perihelion position advances by 5600 arc seconds per century. • All but 43 arc seconds can be accounted for by Newtonian effects and the perturbations of other planets. • Einstein was able to explain those 43 arc seconds via GR. a) Curved space-time for Mercury’s orbit around the Sun. Since Mercury’s orbit is elliptical, its distance from the Sun changes. It therefore passes through regions of different curvature. b) This causes the orbit to precess (amount of shift exaggerated in this figure!)

Bending of Light Einstein said that the warping of spacetime alters the path of light as it passes near the source of a strong gravitational field (i. e. photons follow geodesics). When viewing light from a star, the position of the star will appear different if passing near a massive object (like the Sun). = 4 GM/bc 2 Where is the angle in radians and b is the distance from light beam to object of mass M If b is radius of Sun (7 x 1010 cm), is 8. 5 x 10 -6 rad or 1. 74 arcseconds

Measurements must be made during a solar eclipse, when light from Sun is blocked and stars near the Sun’s edge can be seen. Sir Arthur Eddington headed the attempt to verify Einstein’s prediction during an eclipse in 1919 as the Sun would move in front of a cluster of distant stars. Bent light path also causes a delay in the time for a signal to pass the Sun. This effect has been measured by bouncing radio waves off Mercury and Venus as they pass behind the Sun, and observing signals from solar system space craft. GR effects have been confirmed to an accuracy of 0. 1% using these measurements.

Gravitational Lensing A large galaxy or cluster can act as a gravitational lens light emitted from objects behind the lens display distortion and spherical aberration. Measuring the degree of lensing can be used to calculate the mass of the intervening body. One of the techniques to detect the presence of dark matter. Abell 2218 Light waves passing through areas of different mass density in the gravitational lens are refracted to different degrees. Produces double galaxy images and Einstein Rings (if observer, lens, and source are aligned in a specific way).

Gravitational Redshift A photon’s wavelength is effected by a gravitational field Gravitational potential energy -GMm/r To determine PE for a photon we assign an effective mass based on E=mc 2 m=E/c 2 and since E=hc/ (energy of a photon) m=h/(c ) Conservation of energy for a photon moving from r 1 to r 2 hc/ 1 - GMh/(r 1 c 1) = hc/ 2 - GMh/(r 2 c 2) This gives 2/ 1 = [1 -GM/(r 2 c 2)]/[1 -GM/(r 1 c 2)] Not a rigorous treatment, but a dimensional analysis approximation which agrees with full GR calculation

2/ 1 = ([1 -2 GM/(r 2 c 2)]/[1 -2 GM/(r 1 c 2)])1/2 Full GR calculation agrees with approximation in the limit 2 GM/(rc 2)<<1 If we let r 2 go to infinity and use an approximation for small shifts 2/ 1 = 1 + GM/r 1 c 2 The wavelength shift due to gravitational redshifting is then / = GM/rc 2 What gravitational redshift would be measured for spectral lines originating in the atmosphere of the Sun (in terms of / )? M = 2 x 1033 g, r = 7 x 1010 cm and G = 6. 67 x 10 -8 dyn cm 2/g 2 How about a 1 solar mass compact stellar remnant (white dwarf) with r = 7 x 108 cm? Best cases of measured line shifts due to GR are the white dwarfs Sirius B (3 x 10 -4) and 40 Eridani (6 x 10 -5)

Gravitational Time Dilation t 2/t 1 = ([1 -2 GM/(r 2 c 2)]/[1 -2 GM/(r 1 c 2)])1/2 All clocks run slower in a strong gravitational field than they do in a weaker field. The clock at r 1 will run slower than that at r 2 (r 2 is the position further from the source of gravity and thus experiencing a weaker gravitational effect). Gravitational time dilation has been measured using clocks on airplanes, rockets. When the clocks return, they ran slightly faster (ahead) compared to those on the ground. Let r 2 go to infinity to get T = To/(1 -2 GM/rc 2)1/2 where T is time interval far from mass source On Earth’s surface T = To/(1 -2 g. R/c 2)1/2 Time dilation is about 1 part in 109

http: //hyperphysics. phy-astr. gsu. edu/hbase/relativ/gratim. html#c 5

Gravitational Radiation Just as accelerated charged particles give off EM radiation, GR predicts that certain systems should emit gravitational radiation. Massive objects distort spacetime and a moving mass will produce “ripples” in spacetime which should be observable (e. g. two orbiting or colliding stellar remnants (e. g. neutron stars). LIGO – to detect the ripples in space-time using laser interferometry to measure the time it takes light to travel between suspended mirrors. The space-time ripples cause the distance measured by a light beam to change as the gravitational wave passes by. LISA – NASA’s version in space!

Black Holes Stellar remnants of the highest mass stars (see Chapter 18) The most compact objects in the Universe and therefore represent the most extreme gravitational fields Perfect place to investigate the effects of GR

The escape speed for an object with mass M and size R is 2 GM Vesc = R For the Sun, Radius = 700, 000 km Vesc = 620 km/s What if we squeezed the Sun into 1/4 its current radius? Vesc = 620 x 2 = 1240 km/s What if we squeezed the Sun to ~10 km radius (Neutron star size)? Vesc = 163, 000 km/s ( ~half the speed of light!)

Compressing the Sun further, we would eventually have the escape speed equal to the speed of light. No objects could then escape, including photons Black Hole The critical radius at which the escape speed equals the speed of light is called the Schwarzschild Radius. The sphere around a Black Hole at the Schwarzschild Radius is called the “event horizon, ” because no event inside that sphere can ever be known outside of it.

Schwarzschild worked out the curvature of space-time around a point mass to arrive at the radius where a singularity occurs (some quantity becomes infinite) Rs = 2 GM/c 2 If an object is completely contained within its Rs, a singularity occurs! Recall 2/ 1 = ([1 -2 GM/(r 2 c 2)]/[1 -2 GM/(r 1 c 2)])1/2 If we set r 1 to the Schwarzschild radius, 2 becomes infinite for any r 2. No light can escape from within Rs. Schwarzschild radii for objects with different masses: 1 earth mass: 1 cm 1 solar mass: 3 km Rs = 3 km (M/Msun) 106 solar masses: 3 x 106 km – supermassive BH 109 solar masses: 3 x 109 km – supermassive BH

The average density inside a 1 M blackhole is 1017 g/cm 3 Greater than the density of an atomic nucleus! But, density decreases for more massive BHs = (1 x 1017 g/cm 3)(M/M )-2 Density for 108 M is ~few g/cm 3, not much denser than water. Tidal effects are significant near the Rs gravitational force falls off very quickly with small changes in distance. Since g(r) = GM/r 2 Differentiation yields dg(r)/dr = -2 GM/r 3 …. so tidal forces are most significant at small r What is the difference between the acceleration of gravity at the feet and head of an astronaut just outside a 1 solar mass blackhole?

Strange goings on near a Black Hole. As you get close to a Black Hole, the previous exercise shows that you would get stretched, then torn apart… …because the gravitational pull at your feet is 2 x 1012 cm/s 2 greater than at your head (about 2 billion times gravity on Earth!) Let’s imagine an indestructible astronaut, and give her a clock and a flashlight for her journey to the Black Hole… (We’ll remain behind at a safe distance. )

Strange goings on near a Black Hole. As our astronaut friend approaches the Black Hole, we notice that her flashlight appears redder and redder (to us). When she hovers at a distance very close to the event horizon, the radiation from her flashlight gets gravitationally redshifted even more…. to the infrared and finally radio…. / = GM/(rc 2)

Another effect is seen in the paths of the photons – photons only follow “straight” paths when directed straight up (away) from the blackhole. All other light beams will bend. Only light aimed into the exit cone will escape. At r = 1. 5 Rs, photons aimed horizontally will orbit the blackhole - photon sphere For animations and descriptions of the event horizon and photon sphere see http: //apod. nasa. gov/htmltest/rjn_bht. html

Strange goings on near a Black Hole. Now note that light is like a clock… …electromagnetic oscillations at a given frequency. We see our friend’s oscillations slowing down (due to gravitational redshift). Thus her clock slows down due to gravitational time dilation. At the event horizon, her clock would appear (to us) to stop. We would never see her cross the event horizon…. !

Strange goings on near a Black Hole. What does our astronaut friend see? Her flashlight looks the same to her. Her clock seems to run at the same speed. Looking back at us, she sees… Our flashlight gets bluer. Our clock seems to speed up! T = To/(1 -2 GM/Rc 2)1/2 If the astronaut was your twin sister, after her trip to the Black Hole …you would be older than her! (Your clock really would be running faster than hers)

Strange goings on near a Black Hole. What if our friend continued on through the Event Horizon? She would pass through and not perceive the event horizon in any way. With a supermassive blackhole, even the tidal forces might be survived. There is no physical boundary there but … she could never come back! What would our friend find inside the event horizon? The astronaut would be pulled to the center and crushed down to a point - the singularity What actually happens is not known, because: 1) Current theories are not up to the task. 2) We can never do the experiment!

The observer outside the blackhole can not tell anything about what is going on inside the blackhole. Non-rotating BH The only properties that can be deduced are its mass, electric charge, and angular momentum “blackholes have no hair” In a rotating blackhole, ang mtm is non-zero. The structure differs from non-rotating BH (Kerr found the solutions to Einsteins equations for a rotating BH in 1963). Rotating BH Stationary limit – objects within this limit will be dragged around BH due to rotation. Objects moving at light speed here would appear “stationary” from outside the limit since the reference frame is moving at light speed. Touches EH at poles and stretches to non-rotating EH value. Ergosphere - Energy can be extracted from BH via particles in this region moving on specific trajectories – Penrose process (1969)