The Physics
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Opus in profectus

# General Relativity

## Discussion

### an introduction of sorts for a disorganized section

In the olden days.

I will not define time, space, place and motion, as being well known to all.

Isaac Newton, 1689

Space tells matter how to move. Matter tells space how to curve.

John Archibald Wheeler, 1973 (paid link)

The principle of equivalence

• The absence of a gravitational field (true weightlessness) is indistinguishable from free fall acceleration in a gravitational field (apparent weightlessness).
• Accelerated motion in the absence of a gravitational field (apparent weight) is indistinguishable from unaccelerated motion in the presence of a gravitational field (true weight). The local effects of gravity are the same as those of being in an accelerating reference frame.

Basically…

1. Mass-energy curves space-time — a new version of Hooke's law.
2. Objects trace out world lines that are geodesics (paths of least action in curved space-time) unless acted upon by a net external force — a new version of the law of inertia.

Gravity isn't a force, it's the curvature of space-time caused by the presence of mass-energy.

 ut tensio, sic vis strain ∝ stress space-timecurvature ∝ mass-energystress

The Einstein field equations…

 Rμν − ½Rgμν = 8πG Tμν c4

where…

 Rμν = Ricci tensor curvature R = Ricci scalar curvature gμν = metric tensor Tμν = stress-energy tensor c = speed of light in a vacuum G = universal gravitational constant π = the famous constant from geometry

That's right, I used the plural form — equations. What looks like one equation is actually a set of ten coupled nonlinear partial differential equations. In reverse adjective order these equations are differential because they deal with rates of change (rates of differing), partial because there are multiple variables involved (multiple parts), nonlinear because some of the operations are repeated (a rate of change of a rate of change), and coupled because they cannot be solved separately (every equation has at least one feature found in another).

• Statement of the obvious: Solving these equations turns out to be hard.
• Statement of the awesome: These equations can be broken down into simpler equations by those with a lot of skill. Some of these simpler equations are appropriate to the level of this book, which means you can learn how to do some general relativity. They will be derived with minimal to no proof, however.

### cosmological constant

Space-time is more than just a set of values for identifying events. Space-time is a thing unto itself. The cosmological constant is a quantity used in general relativity to describe some properties of space-time. Here's how it goes.

Maybe gravity is the curvature of space-time caused by the mass-energy of stuff within it plus the energy of space itself.

Rμν − ½Rgμν  =
 8πG Tμ c4
−  Λgμν
space-time
curvature
=  stress from stuff
in space-time
−  stress from empty
space-time itself

Or maybe gravity is the curvature of space-time caused by mass-energy on top of the curvature of space-time itself.

Rμν − ½Rgμν  +  Λgμν  =
 8πG Tμν c4
curvature from stuff
in space-time
+  curvature of
space-time itself
=  mass-energy
stress

Einstein's odd choice of sign might make more sense if you factor out the metric tensor on the left side of the equation. The cosmological constant was invented as a way to hold back gravity so that a static universe wouldn't collapse. (This line of reasoning turns out to be faulty, by the way, but it's a mistake that pays off in the end.)

 Rμν − (½R − Λ)gμν = 8πG Tμν c4

Einstein assumed that the universe was static and unchanging. He thought this was true because that was what astronomers at the time thought they saw when they looked out into their telescopes. A static universe would be unstable if gravity was only attractive. Every piece of matter would attract to every other and any slight imbalance in distribution would would force the whole thing to eventually contract down into itself. Einstein added the cosmological constant to his equations (technically, he subtracted it from the scalar curvature) to hold back gravity so that his equations would have a solution that agreed with the static model.

Write more.

Dark energy is spread absolutely smoothly across the universe.

### unorganized thoughts

• precession of closed (and open) orbits
• In 1859 Urbain Le Verrier (1811–1877) France, director of the Paris Observatory published his observations of an anomaly in mercury's orbit. The precession of mercury's perihelion (point of closest approach to the Sun) had been precessing at 574 seconds of arc per century. Thinking that this was due to the effects of the other planets he calculated the precession rate using Newton's laws at 531 seconds per century, leaving 43 seconds unaccounted for. Can you say "tiny".
• gravitational bending of light
• Confirmed by Arthur Eddington (1882–1944) England in 1919. General relativity replaces Newton's theory of universal gravitation as the most complete theory of gravitation. Newton and Eddington were English. Einstein was German. 1919 was the first year after World War I. Anti-German sentiment was still high in Europe. Eddington's confirmation of Einstein's theory showed that science was above culture and politics. Einstein became a celebrity.
• Einstein cross
• gravitational lensing
• magnification of distant objects
• Gravity Probe A (1976)
• Fly an atomic hydrogen maser on a Scout rocket launched to a height of 10,000 km. A maser is like a laser for microwaves. It produces microwaves of a precise frequency. Measure the doppler shift due to gravity and motion and compare to predicted values (error = 70 ppm = 0.007%)
• Gravity Probe B (2004–2005)
• Tested for frame dragging.

Space never did anything in Newtonian mechanics. Space was just there. In Einstein's theory of relativity, space and time became a thing — a thing that could do stuff like expand, contract, shear, and warp (or bend or curve).

### evolution of the universe

The Friedmann equation (1923). The standard model of cosmology. A single ordinary differential equation that comes out of the ten coupled nonlinear partial differential equations of Einstein.

 1 ⎛⎜⎝ da ⎞2⎟⎠ = ⎛⎜⎝ 8πGρ + Λ ⎞⎟⎠ a2 − k c2 dt 3c2 3

where…

 a = scale factor (size of a characteristic piece of the universe, can be any size) da/dt = rate of change of scale factor (measured by the redshift) ρ = mass-energy density of the universe (matter-radiation density of the universe) k = curvature of the universe (+1 closed, 0 flat, −1 open) Λ = cosmological constant (energy density of space itself, empty space) c = speed of light in a vacuum G = universal gravitational constant π = the famous constant from geometry

Hubble constant, Hubble parameter, expansion rate

 H = da/dt a

The Friedmann equation again.

 1 ⎛⎜⎝ da ⎞2⎟⎠ = ⎛⎜⎝ 8πGρ + Λ ⎞⎟⎠ a2 − k c2 dt 3c2 3
 ⎛⎜⎝ da/dt ⎞2⎟⎠ = ⎛⎜⎝ 8πGρ + Λc2 ⎞⎟⎠ − kc2 a 3 3 a2
 H2 = 8πGρ + Λc2 − kc2 3 3 a2

Critical density.

 ρc = 3H2 8πG

Density parameter.

 Ω = ρ ρc

Big bang. Georges Lemaître.

2nd Friedmann equation.

 1 d2a = − 4πG ⎛⎜⎝ ρ + 3p ⎞⎟⎠ + Λc2 a dt2 3 c2 3

### time dilation

Time runs slower for a moving object than a stationary one. This consequence of Einstein's theory of special relativity is known as time dilation and it works like this…

 t′ = t √(1 − v2/c2)

where…

 t = duration of an event in a moving reference frame t′ = duration of the same event relative to a stationary reference frame v = speed of the moving moving reference frame c = speed of light in a vacuum (a universal, and apparently unchanging constant)

The greater the speed of the moving observer, the closer the ratio v2/c2 is to one, the closer the denominator √(1 − v2/c2) is to zero, the more the time dilates, stretches, enlarges, or expands. From the point of view of a stationary observer, all events in a frame of reference moving at the speed of light take an infinite amount of time to occur. No events can transpire. Nothing can happen. Time ceases to exist.

Time also runs slower in a gravitational field. This is a consequence of Einstein's general theory of relativity and is known as gravitational time dilation. It works like this…

 t′ = t √(1 − 2Vg/c2)

where Vg is the gravitational potential associated with the gravitational field at some location. If you read the section in this book on gravitational potential energy, you may recall that…

 Vg = − Gm r

If you didn't read that section just hear me now when I say that, because of that equation (and ignoring the minus sign), gravitational time dilation works like this…

 t′ = t √(1 − 2Gm/rc2)

where…

 t = duration of an event in the gravitational field of some object (a planet, a sun, a black hole) t′ = duration of the same event when viewed from infinitely far away (a hypothetical location where the gravitational field is zero) m = mass of the gravitating object r = distance from the gravitating object to where the event is occurring (their separation) c = speed of light in a vacuum (a universal, and apparently unchanging constant) G = universal gravitational constant (another universal, and apparently unchanging constant)

This equation says that the closer an event occurs to a gravitating body, the slower time runs; the greater the mass of the gravitating body, the slower time runs; the stronger gravity is, the slower time runs.

For small height changes where the gravitational field is reasonably constant, this approximation works alright.

 t′ ≈ t √(1 − 2g∆h/c2)

And this even more approximate approximation is pretty good too.

 t′ ≈ t ⎛⎜⎝ 1 + g∆h ⎞⎟⎠ c2

where…

 t = duration of an event in the gravitational field of some object (a planet, a sun, a black hole) t′ = duration of the same event when viewed from slightly higher up g = local gravitational field (local acceleration due to gravity) ∆h = height difference between the event and the observer c = speed of light in a vacuum
• Clocks on planes experiment
Prediction Abstract: During October 1971, four cesium beam atomic clocks were flown on regularly scheduled commercial jet flights around the world twice, once eastward and once westward, to test Einstein's theory of relativity with macroscopic clocks. From the actual flight paths of each trip, the theory predicts that the flying clocks, compared with reference clocks at the US Naval Observatory, should have lost 40 ± 23 nanoseconds during the eastward trip, and should have gained 275 ± 21 nanoseconds during the westward trip. Results Abstract: Four cesium beam clocks flown around the world on commercial jet flights during October 1971, once eastward and once westward, recorded directionally dependent time differences which are in good agreement with predictions of conventional relativity theory. Relative to the atomic time scale of the U.S. Naval Observatory, the flying clocks lost 59 ± 10 nanoseconds during the eastward trip and gained 273 ± 7 nanoseconds during the westward trip, where the errors are the corresponding standard deviations. These results provide an unambiguous empirical resolution of the famous clock "paradox" with macroscopic clocks.
• A clock that was raised 33 cm — a third of a meter, a bit higher than a US foot, about two steps up on a typical staircase. Predicted fractional change of 3.6 × 10−17. Measured fractional change (4.1 ± 1.6) × 10−17. It would take about a billion years for that difference to accumulate to one second.
• GPS

### gravitational doppler effect

Motional doppler (special relativity)

 λ = f0 = √ ⎛⎜⎝ 1 + v/c ⎞⎟⎠ λ0 f 1 − v/c

Gravitational doppler (general relativity)

 f = e−∆Vg/c2 f0
 f = 1 − Vg + Vg2 − Vg3 + Vg4 − … f0 c2 2c4 6c6 24c8
 f ≈ 1 − Vg f0 c2
 Δf ≈ ΔVg f0 c2
 f ≈ 1 − Gm f0 c2r
 f ≈ 1 − g∆h f0 c2
• 1959 Harvard Tower Experiment. Pound, Rebka, and Snyder. Jefferson Physical Laboratory, Harvard. Confirmed in an experiment conducted in an elevator(?) shaft at Harvard University by Robert Pound (1919–2010) and Glen Rebka (1931–2015) in 1959. A source of gamma rays was placed at the top of the shaft and a detector at the bottom. The source produced gamma rays of a precise frequency and the detector was designed to detect only gamma rays with that particular frequency. In the process of "falling" down the shaft, the gamma rays were blue shifted to a higher frequency. Pound and Rebka placed the source on a vibrating speaker. When the speaker moved up at the right velocity, the gravitational blue shift was canceled by the motional red shift and the detector would detect the gamma rays. Move with any other velocity and noting is detected. Measure the speed of the source, the local gravitational field, height of detector above emitter, and the speed of light; put numbers into equation; check to see if both sides equal to within the limits of experimental error (~10%, Pound and Snider reduced this to ~1% in 1964).
• 1976 Scout Rocket Experiment. Smithsonian Astrophysical Observatory. The first such experiment was the National Aeronautics and Space Administration/Smithsonian Astrophysical Observatory (NASA-SAO) Rocket Redshift Experiment that took place in June 1976. A hydrogen-maser clock was flown on a rocket to an altitude of about 10,000 km and its frequency compared to a similar clock on the ground. At this height, a clock should run 4.5 parts in 1010 faster than one on the Earth. During two hours of free fall from its maximum height, the rocket transmitted timing pulses from a maser oscillator which acted as a clock and which was compared with a similar clock on the ground. This result confirmed the gravitational time dilation relationship to within 0.01%.

### event horizon

Whatever makes 2Gm/rc2 approach one, makes the dominator √(1 − 2Gm/rc2) approach zero, and makes the time of an event stretch out to infinity. That happens when an event is approaches the following distance from a gravitating body…

 rs = 2Gm c2

This distance is known as the Schwarzschild radius. Another way to write the equation for gravitational time dilation is in terms of this number…

 t′ = t √(1 − rs/r)

The Schwarzschild radius divides space-time into two regions separated by an event horizon. The horizon on the Earth divides the surface of the Earth into two regions — one that can be seen and one that cannot. The event horizon divides space-time up into two regions — an outside where information flows in any direction and an inside where information can flow in but not out. On the Earth, a horizon is associated with an observer. In space-time, an event horizon is associated with a source of extreme gravity.

space time name description
r > rs t′ > t outside time slows down, events at this distance take longer to occur when viewed from locations further outside
r = rs t′ = ∞ event horizon time stops, all events take an infinite amount of time to occur when viewed from outside
r < rs t′ = bi t inside time is mathematically imaginary, time becomes space-like, space becomes time-like (bi is an imaginary number composed of a real coefficient b multiplied by the imaginary unit i where i2 = −1)
r = 0 t′ = 0 singularity time has no meaning, all events happen simultaneously, new physics is needed

Most objects do not have an event horizon. It is a distance that can not exist. All objects that we encounter in our daily lives and most of the objects in the universe are significantly bigger than their Schwarzschild radius. You cannot get so close to the Earth that time would stop. Its Schwarzschild radius is 9 mm, while its actual radius is 6,400 km. Don't think you could stop time by tunneling down to the Earth's core. Gravity within the Earth decreases to zero at its center. You're not closer to the Earth at its center, you're inside it. When you're on the surface of the Earth like you are now, gravity overall pulls you one way — down. If you could go to the center of the Earth, gravity would pull you outward in all directions, which is the same as no direction. Gravity that doesn't pull in any direction can't be strong.

Let's try a bigger object with bigger gravity — the Sun. The Schwarzschild radius of the Sun is 3 km, but its actual radius is 700,000 km. That's not much better. Try the heaviest star known — RMC 136a1. It's 315 times more massive but only 30 times bigger across. Its Schwarzschild radius is 930 km, which is still much smaller than its radius.

The problem (which really isn't a problem) is that the all objects around us and the majority of celestial bodies like planets, moons, asteroids, comets, nebulae, and stars can't be made sufficiently small enough. The sun will die one day and its core will shrink down over billions of years to the size of the Earth, but that's where it will end. The Earth might be blown to smithereens by escaping gas from the dying sun, but it will never be crushed symmetrically into a ball bearing. There essentially is no way to get the Sun's radius to 3 km or the Earth's to 9 mm. RMC 136a1 is a different story, however.

Stars are miasmas of incandescent plasma as the song goes. They're heated from within by the fusion of light elements into heavier ones. That heat keeps them inflated, in a certain sense. When they exhaust their fuel, they lose that heat and start to shrink. For stars like the Sun, hydrogen fuses into helium in the core where pressures are high enough. When all of the core has turned into helium, the star loses the energy needed to keep it pumped up and it starts to shrink.

The sun will shrink until the spaces between atoms are as small as they can get. Such a star is called a white dwarf. Imagine the Sun shrunk down to the size of the Earth. We're still 1000 times or 3 orders of magnitude too big for an event horizon to form.

In the process of shrinking, the Sun will also shed a good portion of its outer layers. That produces a nebulous cloud of incandescent gas surrounding the white dwarf core called a planetary nebula. That's an unfortunate term since it has nothing to directly to do with planetary formation.

Bigger stars have more complicated lifestyles. Some of them can go on extracting nuclear energy by fusing three helium nuclei to form one carbon nucleus. Some will tack additional helium nuclei on to this carbon to form oxygen, neon, magnesium, silicon, sulfur, argon and so on all the way up to iron. Such stars can die in one of two ways. Both involve collapse of the core and the shedding of outer layers. Such a dying star is called a supernova and its a process that happens much more quickly than the death of stars like the Sun — in hours rather than millennia. The remnant core could form a white dwarf if too much of the surface material was ejected, but the more likely outcome is a neutron star or a black hole.

A neutron star is a remnant stellar core with enough mass that its gravitational field is strong enough to overcome electron degeneracy pressure — the quantum mechanical equivalent of the repulsive electrostatic force between electrons. This crushes the orbiting electrons down into the nucleus where they join with protons to form neutrons. Such a star is effectively a giant ball of neutrons. Imagine a stellar core 2 or 3 times the mass of the Sun crushed down to the size of a city, say 10 km in radius. The Schwarzschild radius of a 3 solar mass object is 9 km. We're almost there.

When some really large stars collapse, their remnant cores contain enough mass that gravity will eventually overcome neutron degeneracy pressure — the aspect of the strong nuclear force that keeps neutrons and protons a respectable distance apart. Now there is nothing left to act against gravity and the core crushes itself to zero radius and volume. Not just very small, but actual mathematical zero. Such an object is called a black hole because nothing, not even light, can escape its gravitational hold.

Back to RMC 136a1?

Recall that in the section of this book dealing with gravitational potential energy, that was how the Schwarzschild radius was derived — as the distance from a massive compact object where the escape velocity would equal the speed of light. To this we just added another feature. It's the place where time stops.

### gravitational waves

• binary pulsars spiraling into one another
• suspended aluminum cylinder
• false positive
• discovered for real in 2015, reported in 2016
• interferometer
• LIGO (Laser Interferometer Gravitational Wave Observatory), Advanced LIGO
The Laser Interferometer Gravitational-Wave Observatory (LIGO) is a facility dedicated to the detection of cosmic gravitational waves and the harnessing of these waves for scientific research. It consists of two widely separated installations within the United States — one in Hanford Washington and the other in Livingston, Louisiana — operated in unison as a single observatory