General Relativity

Introduction

GR is Einstein’s theory of gravitation that builds on the geometric concept of space-time 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.

•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 5th dimension(!)•Hard to visualize, but still possible to measure

t t

x x

No gravity gravity

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

See also http://www.pbs.org/wgbh/nova/einstein/relativity/

•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•“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/r2) 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).

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

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).

Gμν = 8πTμν = Rμν – 1/2gμνR

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 space-time 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).

= 4GM/bc2

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 (7x1010cm), is 8.5x10-6 rad or 1.74 arcseconds

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.

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.

Gravitational Lensing

A large galaxy or cluster can act as a gravitational lenslight 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 2218Light 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=mc2

m=E/c2 and since E=hc/ (energy of a photon)m=h/(c)

Conservation of energy for a photon moving from r1 to r2

hc/1 - GMh/(r1c1) = hc/2 - GMh/(r2c2) This gives

2/1 = [1-GM/(r2c2)]/[1-GM/(r1c2)]Not a rigorous treatment, but a dimensional analysis approximation which

agrees with full GR calculation

2/1 = ([1-2GM/(r2c2)]/[1-2GM/(r1c2)])1/2

Full GR calculation agrees with approximation in the limit 2GM/(rc2)<<1

If we let r2 go to infinity and use an approximation for small shifts

2/1 = 1 + GM/r1c2

The wavelength shift due to gravitational redshifting is then

/ = GM/rc2

How about a 1 solar mass compact stellar remnant (white dwarf) with r = 7x108 cm?

Best cases of measured line shifts due to GR are the white dwarfs Sirius B (3x10-4) and 40 Eridani (6x10-5)

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 cm2/g2

Gravitational Time Dilation

t2/t1 = ([1-2GM/(r2c2)]/[1-2GM/(r1c2)])1/2

All clocks run slower in a strong gravitational field than they do in a weaker field. The clock at r1 will run slower than that at r2 (r2 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 r2 go to infinity to get T = To/(1-2GM/rc2)1/2

where T is time interval far from mass source

On Earth’s surface T = To/(1-2gR/c2)1/2

Time dilation is about 1 part in 109

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html#c5

Gravitational RadiationJust as accelerated charged particles give off EM radiation, GR predicts that certain systems should emit gravitational radiation.

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!

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).

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

2GMVesc =

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!)

R

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 radii for objects with different masses:

1 earth mass: 1 cm

1 solar mass: 3 km Rs = 3km (M/Msun)

106 solar masses: 3 x 106 km – supermassive BH

109 solar masses: 3 x 109 km – supermassive BH

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 = 2GM/c2

If an object is completely contained within its Rs, a singularity occurs!

Recall 2/1 = ([1-2GM/(r2c2)]/[1-2GM/(r1c2)])1/2

If we set r1 to the Schwarzschild radius, 2 becomes infinite for any r2. No light can escape from within Rs.

The average density inside a 1 M blackhole is 1017 g/cm3

Greater than the density of an atomic nucleus!

But, density decreases for more massive BHs

= (1 x 1017 g/cm3)(M/M)-2

Density for 108 M is ~few g/cm3, 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/r2

Differentiation yields dg(r)/dr = -2GM/r3

….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 2x1012 cm/s2 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.)

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….

Strange goings on near a Black Hole.

/ = GM/(rc2)

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 = 1.5Rs, photons aimed horizontally will orbit the blackhole - photon sphere

For animations and descriptions of the event horizon and photon sphere seehttp://apod.nasa.gov/htmltest/rjn_bht.html

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!

Strange goings on near a Black Hole.

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)

T = To/(1-2GM/Rc2)1/2

Strange goings on near a Black Hole.

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!

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!

The observer outside the blackhole can not tell anything about what is going on inside the blackhole.

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).

Non-rotating BH

Rotating BHStationary 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)