Einstein's Relativity and Beyond (v1.0)

 

Much of this essay is a summary of classes of Prof Richard Wolfson of Middlebury College among other sources.

Einstein developed the special theory of relativity in 1905, and further expanded it to the general theory in 1915. Special theory is, in its essence, a great equalizer. It says that your frame of reference does not matter - it could be at rest or moving at close to the speed of light. The physics experiments give the same result. You play the same tennis on a steady cruise ship or on land. You eat peanuts the same way on a steady airplane or on land. This simple thing leads to some interesting effects and fundamentally alters classical physics. The general theory of relativity turns out to be a theory of gravity.

After that quantum mechanics was developed. The key concept of quantum mechanics can be illustrated by emptying a jug of water repeatedly in half until you cannot do it anymore. You are left with one molecule of water. Further splitting it completely changes it from water to hydrogen and oxygen atoms and it is no longer water. The key essence of quantum mechanics is similar. You cannot indefinitely keep cutting in half the momentum, energy, the inaccuracy of position, or angular momentum, etc. It is quantized and cannot be split anymore beyond a certain point. I will not dwell into quantum mechanics here because it is explained in depth in two other essays.

The latest developments in physics are particle physics, and theories of everything that unifies everything. I will touch on that in this essay.

These are the four leapfrog advancements in physics. Such leapfrog advances cannot just occur by following the scientific method. There is art, intuition, and a lot of creativity to it. In fact, Einstein just did thought experiments. The actual and many experimental proofs came later.

In this essay, I focus on mostly special, but to a lesser extent general relativity, and on relativity implications for black holes, and touch on particle physics and the theory of everything.

The ancient Greeks thought that the earth was the center of the universe but imperfect, and the heavens were different and perfect with different physics. The natural state of objects on earth is rest while that for the heavens was circular. Copernicus changed it to the sun was the center of the solar system and the planets orbited it. Galileo discovered the moons of Jupiter, and that the sun was not perfect because it had sunspots. So, it became increasingly clear that there is nothing special about earth or “heaven,” and the “heavens” and earth follow the same physics. Newton then came along and formalized his three laws of motion.

All of us were exposed to Newtonian physics in school. The first law (law of inertia) is a body at rest will continue to be at rest and a body in motion will continue to be in motion steadily in a straight line unless a force is applied. The second law is Force = mass * acceleration. The third law is for every action there is an equal and opposite reaction. A side corollary is that the law of motion applies to any frame of reference with uniform motion. This is called Galilean relativity.

The key equations we worked with were:

1. v = u + a * t where v is final velocity, u is initial velocity, a is acceleration and t is time.

2. s = u * t + ½ * a * t **2 where s is distance, u is initial velocity, a is acceleration, and t is time.

3. f = m * a where f is force, m is mass and a is acceleration.

Newton also recognized that the motion of the moon and an apple falling had a commonality. They both exhibited a force on it applied towards the center of the earth. His law of gravity is that all bodies with mass attract each other and F = G * m1 * m2/d **2 where m1and m2 are the masses, G is the gravitational constant and d is the distance between them. To explain planetary motion, newton invented calculus and with it, and his laws of gravity, showed that planetary motion is elliptical around the sun with the sun at one focus of the ellipse The line connecting the Sun to a planet sweeps equal areas in equal times. The square of the orbital period of a planet is proportional to the cube (3rd power) of the mean distance from the Sun. Both bodies attract each other. For the sun and earth, since the sun has so much bigger mass, it is mostly the earth orbiting it. It is similar for moon around the earth.

The concept of force fields naturally arises. The earth has a gravitational field surrounding it. The apple falling towards it only feels its local interaction with the gravitational field. An electrically charged object has an electric field around it that attracts or repels other charged objects in the field. A magnetized object has a magnetic field around it that attracts or repels other magnets in the field. This eliminates the need to assume philosophically difficult things like magical action at a distance. Magnetic fields and electric fields are related. A changing magnetic field generates an electric field, and a changing electric field generates a magnetic field. Maxwell put forth his four laws that formalized this, and then showed with it, how electromagnetic waves are formed by interacting changing electric and changing magnetic fields. He also showed it travels at the speed of light. This includes the whole spectrum from radio to gamma and includes visible light and propagates electromagnetic energy. This was a huge advancement.

Einstein started with the postulate that the laws of physics (including laws of motion and maxwells equations) are the same for any frame of reference in uniform motion. A tennis game on a steady cruise ship will be the same as on land. Eating peanuts on a plane will be the same as eating peanuts on land. Only relative motion matters. This is an extension of gallian relativity by including all of physics including maxwell equations, not just motion. The speed of light is directly derived from maxwell equations, so it will measure the same on any reference frame in steady motion. The speed of light measured on a ship traveling close to the speed of light will produce the same value as someone sitting on earth. The speed of the light source does not change the speed measured. This seems surprising but logically follows. This is it for special theory of relativity!!! Seems intuitively obvious. But the implications are deep for the nature of spacetime itself!! Special relativity is one of the most tested theories in physics and it should be taken as truth.

In 1905 Einstein published 4 papers. One on the Brownian motion established finally that matter consisted of atoms. The second on special theory of relativity is this blog. The third on photovoltaic effect for the first time showed that light behaves as a particle called photon. The fourth laid the foundation on E=mc**2 to come later. The famous equation Energy = mass * c ** 2 was not in the original paper, but an afterthought published in 1907. Einstein was 26 and a clerk in the Swiss patent office!!!

The first consequence is time dilation. Although everything looks normal on reference frame 2 to an observer there, fundamentally time dilates in reference frame 2 moving relative to reference frame 1 from the point of reference frame 1. T2 = T1 * sqrt (1 – (v/c)**2) when v is speed, c is speed of light and t1 and t2 are time. Clocks run slower in the reference frame 2 relative to reference frame 1. This is seen with GPS. Since each satellite is moving relative to your cell phone, the satellite sends its precise time, and it will be slower relative to the cell phone and must be accounted for. The converse is also true – time dilates in reference frame 1 relative to reference frame 2.

One way to test is cosmic rays emitted by the sun and other stars that travel close to the speed of light. One of the particles in it is radioactive with a certain decay rate. The rate of decay has been measured to have slowed down by the exact amount predicted when it reaches here.

The second consequence is distance measures will contract in reference frame 2 relative to reference frame 1 although everything looks normal on reference frame 2 to an observer there. s1 = s2 * sqrt (1 – (v/c)**2). There are two thought experiments on how strange this is. If one twin goes in a spaceship to a distant star and returns in 20 years earth time, she would have aged much less than her twin who stayed behind!!

Another example is if the twin goes to a galaxy 1 million light years away, to her the journey is much shorter if she is traveling fast enough and it takes much less time, and she could have aged just a small amount on return, but earth would be two million years in the future!! Both time and space are not absolute, but relative.

How can time dilate, and distance contract and clocks run slower for both reference frames from the others perspective? It sounds strange but true none the less. The proof is not included. The twin in the spaceship from her perspective can say that earth went at a rapid pace to a faraway place and came back. So, the twin on earth should be much younger than her when she returns. The subtle point is that the earth is more or less in uniform motion, but the space craft is not. Later 0n, the special theory is generalized into the general theory by allowing the reference frame to be in a force field and there is no uniform motion.

Another consequence is that simultaneous events on one frame may not be simultaneous on another frame. In fact, even the order of the events may be reversed. Does that mean you can die before being born? Does causality break down? The answer it turns out is that that can happen only if the two events are far enough so even light cannot go from one to the other in time. To understand this, consider the mars rover runs into a problem. It takes 11 minutes for the signal to reach earth and another 11 minutes for the recovery instruction to be beamed back. There can be no causally linked events between the two in the 22 minutes. That period is not the past nor the future. It is not anywhere. The redefined definition of past is events that can influence the present, and redefined definition of future is events that can be influenced by the present. So, causality is not violated.

Another consequence is that nothing – matter, energy or information can go faster than the speed of light. One way to look at it to partially explain this is that like mass, energy has inertia. As the particle gets closer and closer to the speed of light, its energy gets enormous, and its inertia to go faster gets higher and higher and at the speed of light it goes to infinite. Another way the mathematician Minkowski explained it, is by looking at Einstein’s equations. If you plot time versus the space in 4 dimensions (the space dimensions are Euclidian while the time dimension is not, called Minkowski spacetime which is fundamental to understanding Maxwells equations, as well as Einsteins equations), everything no matter what has a vector whose length is the speed of light – just the axis’s are rotated for different frames of references. For a particle going in a straight line at the speed of light, the vector has zero component in the time axis because time dilation goes to infinite and time stops. For an object at rest, it is pointing in the time direction because it is not moving along any space direction. So, every object no matter what is going at the speed of light in spacetime. I hope I captured the argument correctly!! So, you cannot go faster than light. Interesting explanation!!

Another consequence is that if a spaceship travels at 0.8 times the speed of light and launches a smaller spaceship that travels 0.8 times the speed of light relative to the big spaceship, it will end up traveling 0.97 times the speed of light relative to earth, not 1.6 times the speed of light.

The bottom line is intuition and common sense is based on limited experience. Even though special relativity challenges common sense, it is the absolute truth.

In Newtonian physics, it was deemed a coincidence that the inertial mass to push and accelerate say a ball on a flat surface was the same as the gravitational mass that gravity exerts to accelerate the ball. Einstein made it a centerpiece of general relativity that they are indistinguishable. Therefore, general relativity is also about a theory of gravity.

That means an accelerating spaceship in outer space is indistinguishable from an elevator at rest at the base due to gravity. Also, an elevator in free fall is indistinguishable from a spaceship in uniform motion in outer space. So, gravity can be transformed away. So, gravity cannot real!! But something causes masses to attract. So, what is it? Einstein proposed a four-dimensional geometry of space time that is curved, and the curvature is gravity. It is not Euclidian geometry. it is not a geometry of parallel lines that never meet. Spacetime is curved by the presence of matter and the presence of energy. Matter, that is not under the influence of any forces (does not include gravity because gravity is just curved space time), will respond to the curved space time by moving in the straightest possible course through that curved space time. This is very hard to grasp. The mathematics is extremely complex. The general theory of relativity must be one of the crowning intellectual achievements in history. It generalized special relativity by removing the limitation of frame of reference being only in uniform motion and explained gravity. I will not even attempt to try and give an analogy to illustrate!!! The best I can do is spacetime is like a trampoline that is warped by a mass or energy blob on it. This creates a depression, and the motion of other particles near it can be explained by this depression. The theory was launched full born with no experimental evidence!! And its predictions have come true!!

For predictions where gravity is weak (escape velocity much less than speed of light), general relativity only slightly differs from Newtonian physics. But where it is very strong it deviates.

One prediction is that unlike Newtonian physics that planetary orbits are perfect ellipses that repeats, Einstein predicts that on every rotation it is shift slightly. This is called orbital precession. This was confirmed with mercury and also two pulsars (spinning neutron stars) rotating around each other.

Another prediction is that time should run slower where spacetime is very sharply curved. Example a clock on the ground will run slower than one in a satellite. This is verified. This is also something that is dealt with in GPS where you have a satellite in orbit and your GPS receiver – the cell phone.

Another prediction is that light should be bent by gravity. This is called gravitational lensing. This has been confirmed.

According to general relativity, when two black holes spiral in towards each other and merge, the resulting gravitational-wave signal – or general relativity waveform – has a characteristic profile that, as time passes, increases in frequency and amplitude until they peak. A very recent discovery and confirmation is that there is a continuous background of such gravity waves that constantly ripple through space-time. This was proven by observing pulsars, (which pulse with almost the precision of an atomic clock, and noting minute changes in frequency over years that can only be due to such a background of gravity waves. Next step is to evaluate different types of gravity waves and try to assign specific sources for specific ones.

Black holes are thought to form when a large enough star (size predicted by relativity) explodes in a super nova explosion. A lot of matter is ejected and forms the cradle for new stars. What is left collapses to a small size with a huge gravity. When the gravity is large enough to make the escape velocity from it greater than the speed of light, a black hole forms because nothing – not even light can escape. For earth, the escape velocity is 7 miles/sec. The sphere around the black hole where the escape velocity equals the speed of light is called the event horizon. If a spaceship free falls towards the event horizon, time gets dilated due to relativity as seen by an observer outside and at the event horizon the dilation is infinity, and the observer will see the ship approach the event horizon infinitely slowly. However, for the person in the spaceship heading to the black hole, everything is normal, and you eventually cross the black hole event horizon, but the tidal forces of gravity like different gravity at different points by the moon on earth will stretch you and compress you sidewards and you are in trouble.

Scientists have been bombarding atomic nucleus with higher and higher energy particles and a plethora of even smaller particles have been created as a result. Now these have been categorized into a standard model of particles and forces that tabulates their mass, energy, spin, charge, etc. Protons and Neutrons are not elementary and made up of other particles. They consist of quarks which are unusual in that they carry a fractional electric charge (third or two third of an electron charge). Protons have two up quarks with 2/3 charge and one down quark with 1/3 charge. Neutrons have two down quarks with 1/3 charge and one up quark with 2/3 charge. Ordinary matter appears to be made up of just a few fundamental particles – electrons, up quark and down quark and electron neutrinos. But several combinations of quarks are possible that potentially leads to additional particles, but they do not appear to be stable. Neutrons are unstable when taken out of the atom (decay rate of a about 20 minutes), but protons are stable. There are also anti particles like anti quarks. Scientist now believe there are only three fundamental forces. Gravity (but Einstein treats it as curvature of space time so is it really a force?), electro weak force (where electricity, magnetism and weak nuclear force are aspects of it), and color (which is the force binding quarks to create hadrons and mesons). An aspect of the color force keeps the nucleus together. These forces fall off with distance. Color is extremely strong and operates at very short distances. So cannot pull apart quarks, and there are no isolated quarks. We know this because beyond a certain size, atomic nuclei are unstable.

The huge advance to date is not completely satisfactory because there is still no theory of everything. Quantum mechanics and special relativity and electroweak force has been harmonized in quantum electrodynamics. Quantum thermodynamics merges the fields of thermodynamics and quantum mechanics. The unification of the color force and electroweak force and finally the unification of gravity with the other forces is needed. Is there something even more fundamental that explains the various fundamental particles in the standard model? Quantum Mechanics and general relativity has also not been harmonized. We need to understand quantum fluctuations in spacetime with a theory on quantum gravity. We need to understand what a graviton particle is which embodies the quantum particle for gravity. Normally relativity deals with the very large and quantum mechanics deals with the very small. These converge very close to the center of the black hole and immediately after the big bank in the plank epoch. I will explore the state of advances in this area in a separate essay.