15.1 Tenets of General Relativity 15.2 Tests of General Relativity 15.3 Gravitational Waves

1. CHAPTER 15General Relativity • 15.1 Tenets of General Relativity • 15.2 Tests of General Relativity • 15.3 Gravitational Waves • 15.4 Black Holes • 15.5 Frame Dragging An excellent introductory General Relativity text Albert Einstein (1879-1955) There is nothing in the world except empty, curved space. Matter, charge, electromagnetism, and other fields are only manifestations of the curvature. - John Archibald Wheeler

2. Inertial Mass and Gravitational Mass Recall from Newton’s 2nd Law that an object accelerates in reaction to a force according to its inertial mass, mi: Inertial mass measures how strongly an object resists a change in its motion. Gravitational mass, mg, measures how strongly it attracts and is attracted by other objects: Equating the forces, we get a ratio of masses: We always take the inertial and gravitational masses to be equal. Einstein considered this equivalence fundamental. where:

3. 15.1: Tenets of General Relativity General relativity is the extension of special relativity to non-inertial (accelerating) frames. And because the effects of gravity and acceleration prove to be indistinguishable, it will also be a theory of gravity. It’s based on two concepts: (1) the principle of equivalence—the extension of Einstein’s first postulate of special relativity to the case of non-inertial reference frames. (2) the modeling of these effects as the curvature of space- time due to matter.

4. Next, we’ll ask what happens when the rocket takes off and accelerates through empty space. Principle of Equivalence The principle of equivalence is an experiment in non-inertial reference frames. Consider an astronaut sitting in a confined space on a rocket placed on Earth. The astronaut sits on a scale that indicates a mass M. The astronaut also drops a safety manual that falls to the floor.

5. Principle of Equivalence Now let the rocket accelerate through space, where grav-ity is negligible. If the acceleration is g, then the scale indicates the same mass M that it did on Earth, and the safety manual still falls with the same acceleration as measured on earth. The question is: How can the astronaut tell whether the rocket is at rest on earth or accelerating in space? Principle of equivalence: There is no experiment that can be done in a small confined space that can detect the difference between a uniform gravitational field and an equivalent uniform acceleration.

6. Light Deflection • Consider accelerating where gravity is negligible, with a window to allow a beam of starlight to enter the spacecraft. Since the velocity of light is finite, it takes time for the light to reach the spaceship’s opposite wall. • During this time, the rocket has accelerated upward. Inside the rocket, the light path bends downward. • The principle of equivalence implies that, at rest on earth, light must also bend downward.

7. Curvature of Space and Geodesics • Light’s constant velocity in special relativity implies that it travels in a straight line. Light bending for the Earth observer seems to violate this premise. • Einstein recognized that we need to expand the notion of a straight line. • The shortest distance between two points on a flat surface differs from the shortest distance between points on a sphere, which is curved. We shall expand our definition of a straight line to include any minimized distance between two points. Such a minimal distance on a curved surface iscalled a Geodesic. New York Paris

8. Curvature of Space Thus if the space-time near a massive body is not flat, then the straight line path of light (and other objects) near that body will appear curved.

9. The Unification of Mass and Space-time • Einstein postulated that the mass of the Earth creates a dimple on the space-time surface. In other words, the mass changes the geometry of space-time. • The geometry of space-time then tells matter how to move. Gravity is no longer a force, but, instead, a property (curvature) of space-time. • Einstein’s famous field equations sum up this relationship as: Matter tells space-time how to curve. Space-time curvature tells matter how to move.

10. Recall the Space-time Interval and Metric Recall the space-time interval, Ds. Ds2 = Dx2 + Dy2 + Dz2 – c2Dt2 This interval can be written in terms of the space-time metric:

11. 0 1 2 3 The space-time metric and summation notation We can rewrite the expression for the space-time interval: where: and When the same index appears as a superscript and a subscript, summation is assumed. This is called Einstein Summation Notation: Use of Greek letters means 0, 1, 2, 3

12. In General Relativity, space is curved, and the space-time metric can be more complex. This more general metric coefficients of general relativity, which may not be -1’s, 0’s, and 1’s, are denoted by gmn: Example: An expanding (flat) universe Ds2 = a(t)2 [Dx2 + Dy2 + Dz2] – c2Dt2 where a(t) ~ t q Values of q range from 1/2 (in a radiation-dominated universe) to 2/3 (in a matter-dominated universe).

13. The Mathematics of General Relativity: Tensors A Tensor is a function of one or more vectors that yields a real number. gmn is a Tensor. It takes two input vectors and yields a number, the interval: Because gmn operates on two vectors, we say it’s a tensor of Rank 2. Vectors can undergo dot products with one other vector to yield a number, so they’re tensors of Rank 1. Scalars have Rank zero. The rank is also the number of indices on the tensor and the dimension of the matrix necessary to write it down.

14. The Mathematics of General Relativity General Relativity distinguishes between vectors and tensors that are covariant (with lower indices) and contravariant (with upper indices). To raise or lower an index, simply multiply by the metric: Ordinarily, we don’t usually have to worry about this because our metric is simple, and covariant and contravariant tensors are essentially the same. To raise the indices of the metric gmn itself, just take its inverse:

15. The Geodesic Equation In Newtonian space, geodesics are straight lines, and one way of saying this is that acceleration is zero: where t is proper time, and xa is the position vs. t of a test particle. In curved space, this expression generalizes to: where is called a Christoffel symbol and is given by:

16. The Ricci Curvature The curvature of space-time is complicated because there are several dimensions, and the curvature at each point can be different in each dimension (including time). Think of a saddle in two dimensions: The curvature of space-time is given by the Ricci Tensor:

17. The Einstein tensor, Gmn is a related measure of the curvature. The Einstein tensor can be written in terms of the Ricci tensor: where R is the trace of the Ricci tensor.

18. Matter’s effect on space-time occurs through the stress-energy tensor, T. Tttis the mass-energy density Txt , Tyt and Tzt are how fast the matter is moving—its momentum Txx , Tyy and Tzz are the pressures in each of the three directions Txy , Txz and Tyz are the stresses in the matter

19. Einstein’s Field Equations This set of coupled nonlinear partial differentialequations (one for each element) relates the curvature of space, Gmn, to the energy-momentum tensor, Tmn : Only six component equations are independent. and where G is the usual gravitational constant. The goal is to solve for gmn, for all values of m and n. In free space, where Tmn = 0, this reduces to: The correspondence principle: Einstein’s Field Equations reduce to Newton's law of gravity in the weak-field and slow-motion limit. In fact, the above constant (8p G / c4) is determined this way.

20. The spherically symmetrical case: the Schwarzschild Solution Using spherical coordinates, r, q, f, and spherical symmetry, we can solve Einstein’s Field Equations for the metric to find: The other elements of gmn are zero, and Note that, when r = 2GM/c2 (called the Schwarzchild radius), this becomes: What’s going on?

21. 15.4: Black Holes

22. Escape velocity and black holes An object will escape from a massive body when its kinetic energy equals or exceeds its gravitational potential energy: Consider the case that the escape velocity is the speed of light: Yeah, we probably shouldn’t use the non-relativistic expression for the kinetic energy here… This can only occur when the mass M is crammed into a radius r: The Schwarzschild radius!

23. Black Holes and Event Horizons • When a star’s thermonuclear fuel is depleted, no heat is left to counteract the force of gravity, which becomes dominant. The star’s mass collapses into an incredibly dense ball that could warp space-time enough to not allow light to escape. The point at the center is called a singularity. A collapsing star greater than 3 solar masses will distort space-time in this way to create a black hole. Karl Schwarzschild determined the radius of a black hole, known as the event horizon:

24. e- e+ e- Hawking Radiation Event horizon • Due to quantum fluctuations, particle-antiparticle pairs are created near the event horizon. One particle falls into the singularity as the other escapes. Antiparticles that escape radiate as they annihilate with matter. Black hole Hawking calculated the blackbody temperature of the black hole to be: k is Boltzmann’s constant The power radiated is: s is the Stefan-Boltzmann constant from blackbody theory.

25. Black Hole Evaporation Small primordial black holes (10-19 solar masses) could be detected by their Hawking radiation, but it is negligible for other black holes. • Energy expended to pair production at the event horizon decreases the total mass-energy of the black hole, causing the black hole to slowly evaporate with a lifetime: Stephen Hawking n is a constant = 1 / 15,360p The smaller the mass the shorter the lifetime. Solar-mass black holes would live much longer than the age of the universe, but small black holes (<1014 g, about the size of a mountain) would explode.

26. Black Hole Detection • Since light can’t escape, black holes must be detected indirectly. Mass falling into a black hole would emit x rays:

27. Black Hole Candidates • Several plausible candidates: • Cygnus X-1 is an x-ray emitter and part of a binary system. It’s roughly 7 solar masses. • The galactic center of M87 is 3 billion solar masses. • That of NGC 4261 is a billion solar masses.

28. 15.2: Tests of General Relativity • Bending of Light • A total solar eclipse allowed viewing starlight passing close to the sun in 1919. It was bent, causing the star to appear displaced. • Einstein’s theory predicted a deflection of 1.75 seconds of arc, and two measurements found 1.98 ± 0.16 and 1.61 ± 0.40 seconds. • Many more experiments, using starlight and radio waves from quasars, have confirmed Einstein’s predictions about the bending of light with increasing accuracy.

29. Gravitational Lensing When light from a distant object like a quasar passes by a nearby galaxy on its way to us on Earth, the light can be bent to give multiple images as it passes in different directions around the galaxy.

30. Gravitational Red-shift • Another test of general relativity is the predicted frequency change of light near a massive object. • A light pulse traveling vertically upward from the surface of a massive body will gain potential energy and lose kinetic energy, as with a rock thrown straight up. • Since a light pulse’s energy depends on its frequency n through E = hn, as the light pulse travels vertically, its frequency decreases. • This phenomenon is called gravitational red-shift.

31. Gravitational Red-shift Experiments • An experiment conducted in a tall tower measured the blue shift in frequency of a light pulse sent down the tower. The energy gained when traveling downward a distance H is mgH. Ifnis the frequency of light at the top and n’ is the frequency at the bottom, energy conservation gives hn = hn’ + mgH. • The effective mass of light is m = E / c2 = hn / c2. • This yields the relative frequency shift: • Or in general: • Using gamma rays, the frequency ratio was observed to be:

32. Gravitational Time Dilation • The frequency of a clock decreases near a massive body, so a clock in a gravitational field runs more slowly. • A very accurate experiment was done by comparing the frequency of an atomic clock flown on a Scout D rocket to an altitude of 10,000 km with the frequency of a similar clock on the ground. The measurement agreed with Einstein’s general relativity theory to within 0.02%.

33. Perihelion Shift of Mercury • The orbits of the planets are ellipses, and the point closest to the sun is called the perihelion. It has been known for hundreds of years that Mercury’s orbit precesses about the sun. After accounting for the perturbations of the other planets, 43 seconds of arc per century remained unexplained by classical physics. The curvature of space-time explained by general relativity accounted for the 43 seconds of arc shift in the orbit of Mercury.

34. Light Retardation • The path of the light passing by a massive object is longer due to the space-time curvature, causing a delay for a light pulse traveling close to the sun. • This effect was measured by sending a radar wave to Venus, where it was reflected back to Earth near the sun. It saw a delay of about 200 ms, in excellent agreement with the general theory.

35. 15.3: Gravitational Waves • When a charge accelerates, the electric field surrounding the charge redistributes itself. This change in the electric field produces an electromagnetic wave, which is easily detected. In much the same way, an accelerated mass should also produce gravitational waves. • Gravitational waves carry energy and momentum, travel at the speed of light, and are characterized by frequency and wavelength. • As gravitational waves pass through spacetime, they cause small ripples. The stretching and shrinking is on the order of 1 part in 1021 even due to a strong gravitational wave source. • Due to their small magnitude, gravitational waves would be difficult to detect. Large astronomical events could create measurable spacetime waves such as the collapse of a neutron star, a black hole or the Big Bang. • This effect has been likened to noticing a single grain of sand added to all the beaches of Long Island, New York.

36. Gravitational Wave Experiments • Taylor and Hulse discovered a binary system of two neutron stars that lose energy due to gravitational waves that agrees with the predictions of general relativity. • LIGO is a large Michelson interferometer device that uses four test masses on two arms of the interferometer. The device will detect changes in length of the arms due to a passing wave. NASA and the European Space Agency (ESA) are jointly developing a space-based probe called the Laser Interferometer Space Antenna (LISA) which will measure fluctuations in its triangular shape.

37. 15.5: Frame Dragging • Josef Lense and Hans Thirring proposedin 1918 that a rotat-ing body can literally drag space-time around with it as it rotates. This effect is called frame dragging. • It was observed in 1997 by noticing fluctuating x rays from several rotating black holes. The objects were precessing from the space-time dragging along with it. • Also, planes of satellite orbits shift 2 m per year in the direction of the Earth’s rotation—in agreement with the predictions of the theory. • Global Positioning Systems (GPS) have to utilize relativistic corrections for the precise atomic clocks on the satellites.

38. Mach’s Principle So what causes inertial mass, and what determines which frames are inertial and which are not? Mach speculated that inertial frames are those that have constant velocity with respect to the “fixed stars,” that is, the rest of the universe. Ernst Mach (1838-1916)

39. Gödel’s Rotating Universe Einstein acknowledged Mach’s ideas in his writings on GR, but GR actually violates it! In Gödel’s rotating universe, GR yields different trajectories with respect to the fixed stars than it does in a stationary universe.