His key insight was that acceleration is indistinguishable from gravity. We can see this clearly in the case of astronauts orbiting Earth. We typically talk about them being in “zero-gee” (no gravity), when in fact they are still very much within Earth’s gravitational field. They do not perceive the force of gravity because they are in free fall, along with their entire spacecraft. According to Einstein’s theory of “general relativity,” there is no observable difference between free fall in a gravitational field and constant-velocity motion in a part of space with no gravitational field.
Mathematically, general relativity is quite a bit more difficult than special relativity. You can get a general idea of this by looking at Einstein’s field equations, which replace Newton’s law of gravitation:
The indices μ and ν refer to the four coordinates of spacetime, and each pair of indices (00, 01, 02, 03, 11, 12, 13, 22, 23, and 33) corresponds to a different equation. Thus the line above actually comprises ten separate equations.
The left side of Einstein’s field equations measures the curvature of space, and the right side, the “stress-energy tensor,” represents the propagation of matter and energy (which are equivalent!). John Wheeler, a leading relativity theorist, expressed the meaning of this equation succinctly: “Matter tells spacetime how to curve, and curved space tells matter how to move.” The above equation led to the discovery of black holes and to the Big Bang theory, and on a more mundane level it provides additional correction terms to GPS satellites. In fact, the general relativity corrections to GPS are larger than the special relativity corrections.
It was also the general theory of relativity that led to Einstein’s prediction of the curving of light rays in a gravitational field. For example, the light from distant stars bends as it passes by the Sun. When this prediction was confirmed by measurements taken during the solar eclipse of 1919, Einstein rocketed to sudden celebrity.
Above A page from Albert Einstein’s General Theory of Relativity. He donated the complete original manuscript of his ground-breaking theory to the Israeli Academy of Sciences and Humanities.
But now let’s answer the question posed earlier: How did Einstein realize that matter and energy are equivalent? By 1915, when he wrote down the equations of general relativity, the equivalence was second nature to him. But in 1905, when he was still just a patent clerk, general relativity was not even a gleam in his eye yet. All he had to work with was special relativity.
EINSTEIN DISCOVERED HIS MOST famous equation by pursuing a seemingly innocuous observation to its logical conclusion. He asked what would happen if a body emitted two photons in opposite directions, and if it was viewed in two different inertial frames: one at rest with the body, the other moving at velocity v, perpendicular to the photons. He showed that the photons would be blue-shifted (have higher frequency) in the moving coordinates. Thus, because of his first equation E = hν, they must also have higher energy. Einstein argued that the energy could only have come from the kinetic energy of the body that emitted them. The Newtonian formula for the kinetic energy is 1/2 mv2, half the mass times the velocity squared. But the velocity of the object in the moving coordinate frame could not have changed when it emitted the photons, because their momenta cancel each other. Therefore the mass must have changed! The body has converted mass to energy, and the amount of mass converted can be computed:
The most famous equation in history follows as a consequence.
Einstein’s paper, “Does the Inertia of a Body Depend Upon Its Energy Content?” is both beautiful and horrifying to a mathematical purist. It is only three pages long. It is wonderful to see how the strands of Einstein’s thought weave together, combining the light-quanta hypothesis with special relativity like two instruments in a duet. But the “lazy dog” of a composer, the enfant terrible who did not care about his mathematics courses, is still very much in evidence. Einstein does not actually prove that E = mc2 ! He makes an approximation at one point, and therefore he proves only that E ≈ mc 2 (that is, energy is approximately equivalent to matter). He makes no real attempt to determine how accurate the approximation is. It’s as if he couldn’t be bothered. Why spoil a “funny and infectious” thought with a pedantic mathematical proof? Later, of course, Einstein and others would go back and provide more rigorous arguments for this most important physical principle.
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from a bad cigar to westminster abbey dirac’s formula
By 1922, Albert Einstein was an international celebrity, mostly because of his theory of general relativity. Meanwhile, the quantum revolution that he had begun was continuing to progress apace, mostly without Einstein’s participation. The world of physics was in turmoil, with just as many skeptics as believers in the new quantum physics. Even the believers were not sure just how much to trust their new theories.
What is quantum physics, and what is it that makes it so revolutionary? At its most basic level, it simply asserts that the measurements that physicists make, such as energy, electric charge, and angular momentum, are quantized. They are not infinitely divisible; there is a smallest unit of energy, of charge, etc.
Taken by itself, that statement may seem interesting but hardly revolutionary. The revolutionary implications become apparent when you start prying into the details. Individual quanta do not behave like anything else we are used to in the macroscopic world. For example, Einstein showed that a photon is both a particle and a wave. How is that possible? Our intuition, adapted to a universe where particles are particles and waves are waves, is helpless to explain it. At that point, mathematics becomes our only guide.
ψ denotes a wave function, which represents (for example) the state of an electron. E represents the electron’s energy, m its mass, and p its momentum. Both α and β are spin matrices or “spinors.” Dirac’s equation modifies Einstein’s to say that the energy of a particle depends on its mass, momentum, and spin.
Another prediction of quantum theory that seemed to require particles to perform impossible feats was the quantization of angular momentum.
At that time, it was believed that for a particle of mass m, the quantum of spin is mh/2π. (Here, h is Planck’s constant, which also appeared in the formula for the energy of a quantum of light.) If you measure the angular momentum of the particle about any axis, you will get a multiple of this same quantum. Such a phenomenon would be completely impossible in classical physics. A classical particle, such as a planet or a bowling ball, has a pre-existing axis of rotation before you measure it. That axis may be askew from the direction you choose to measure. If so, you will only succeed in measuring part of the angular momentum.
But for a quantum particle, any observation of the angular momentum is an “all or nothing” proposition. Either you will see all of the angular momentum about the axis that you choose, or none of it. It is almost as if the particle waits for you to observe it, and then at that instant “decides” whether to spin around that axis or not. This “observer effect,” in which the observer seems to affect the system being observed, is ubiquitous in quantum physics; remember, for instance, that a photon seems to decide whether it is a particle or a wave based on what experiment the observer chooses to perform.
Two physicists in Frankfurt, Germany, saw an excellent opportunity to test, and possibly refute, the quantum theory. Otto Stern and Walther Gerlach had developed a method for producing a beam of silver atoms. When a magnetic field was applied to the beam, according to the quantum theory, the atoms in the beam would be directed right or left, depending on their axis of rotation. If they were rotating counterclockwise about the axis of the magnetic field (“spin-up”) they would be deflected one way. If they were rotating clockwise (“spin-down”) they would be deflected the other way. Thus the beam would split in two.
Above A digital interpretation of quantum particles.
However, if the world were described by classical physics, the spin directions of the silver atoms would be randomly oriented. Some atoms would be deflected a lit
tle bit in one direction, and some would be deflected a little bit in the opposite direction, and all would be deflected by different amounts. Instead of splitting into two beams, the silver atoms would fan out into a diffuse, wider beam. Which theory would be proven right?
Unfortunately, when they first looked at their collector plate, Stern and Gerlach did not see anything! Their beam was too weak, and the number of silver atoms deposited on the plate was too small to detect.
But as Stern hunched over the plate, with Gerlach peering over his shoulder, they saw two dark lines magically appear where none were visible before. The reason, Stern later deduced, was that both he and Gerlach were cigar smokers. “My salary was too low to afford good cigars, so I smoked bad cigars,” he wrote. “These had a lot of sulfur in them, so my breath on the plate turned the silver into silver sulfide, which is jet black, so easily visible. It was like developing a photographic film.” (A re-enactment in 2003 showed that “cigar breath” is not strong enough to produce the effect Stern described. However, exposing the silver directly to cigar smoke does work, and presumably that is what happened.)
Thus, thanks to a cigar, the quantum prediction was confirmed: the beam split in two. However, this was not the end of the story. With hindsight, physicists now know that the effect Stern and Gerlach had observed was not what they had been looking for. Like Columbus, who went looking for India but found America instead, they had had gone looking for the orbital angular momentum of the electrons rotating about the silver atom’s nucleus. What they found instead (without realizing it) was the spin of the electrons themselves. The discovery had repercussions no one could have expected.
SOME PHYSICISTS had considered the possibility that electrons could spin. However, in order to achieve an angular momentum of mh/2π, the electron would have to spin so fast that its outer surface would be traveling faster than the speed of light! Of course, according to the theory of relativity that was not possible.
A young graduate of Cambridge University, Paul Adrien Maurice Dirac, set out in 1927 to reconcile the quantum mechanics of the electron with special relativity. He started with an equation that Einstein himself had written down:
This may look somewhat familiar; it is the formula for the equivalence of matter and energy (E = mc2), only it has been corrected to include the momentum of the electron (p). Another immediately obvious change is that the formula now gives the square of the energy, rather than the energy itself. Dirac was convinced this was a defect, and he looked for a way to take the square root of the equation. However, simply writing a square root in front was not acceptable to him. He had a highly aesthetic approach to physics, and many times said that the equations of physics must be beautiful. Square roots, to Dirac, were ugly.
Instead, Dirac wrote the formula for the electron in the following way:
where p1, p2, p3 represent the three components of the electron’s momentum in three-space. The mysterious quantities α1, α2, α3 and β satisfy the following relations:
I have displayed these formulas to make a point: they are virtually identical to the quaternion formulas that William Rowan Hamilton had written 80 years earlier! Only the names have changed (and –1 has been changed to 1 in the first equation). Dirac had, in a sense, rediscovered quaternions, although he wrote them as 4-by-4 matrices.
The other change that Dirac made to Einstein’s formula was to rewrite the energy as an operator on a wave function, Ψ. This is consistent with the philosophy of quantum mechanics: any observable quantity of a particle is not merely a number but an actual physical operation applied to that particle. (This is why the observer is such an intrinsic part of quantum mechanics.) Thus, the final form of Dirac’s equation looks like this:
Here I have, for convenience, condensed the three alpha matrices into one symbol α, and written the momentum as a single vector p. In order for the formula to make mathematical sense, the wave function ψ has to be a quaternion-like object with four components. This was actually the most puzzling aspect of the equation to physicists, for two reasons.
Firstly, it had four components instead of two. Physicists could make sense of two components—they would represent the spin-up and spin-down states of an electron. But what was the meaning of the other two?
Secondly, the wave function did not behave like a vector (an “arrow” in spacetime). When you rotate space by 360 degrees, the wave function rotates by only 180 degrees, and thus the electron goes from “spin-up” to “spin-down.”
Above Paul Dirac standing in front of a blackboard displaying a quantum mechanical model of the hydrogen molecule.
The second point shows that electrons are not like bowling balls or planets. However, there is an ingenious analogy that goes by the name of the “Feynman plate trick” or “Dirac belt trick.” Place a plate in your open palm in front of you. Now rotate the plate 360 degrees, by rotating your arm in a circle while keeping your palm up. You will find your arm is in quite an awkward position; unlike the plate, it has not come back to its original state. It has rotated 180 degrees. But if you continue and rotate the plate one more time, your arm will come back to its normal, comfortable position! The whole system of “arm plus plate” behaves like a quaternion.
Although Einstein’s formula E = mc2 may be better known to the public, Dirac’s formula may well be of greater significance both to physicists and mathematicians. “Of all the equations of physics, perhaps the most ‘magical’ is the Dirac equation,” wrote Frank Wilczek of MIT in 2002, on the centennial anniversary of Dirac’s birth. “It is the most freely invented, the least conditioned by experiment, the one with the strangest and most startling consequences … [It] became the fulcrum on which fundamental physics pivoted.”
WHY DID IT CHANGE physics so much? Let’s start with those two extra components of the electron wave function. Dirac explained them as particles with negative energy, or “holes” in space. They should appear to be particles just like electrons, but with a positive charge. He proposed the idea in 1931, with great hesitancy. Other physicists ridiculed the idea. Wolfgang Heisenberg wrote, “The saddest chapter of modern physics is and remains the Dirac theory.”
Yet within a year, Carl Anderson of Caltech had discovered Dirac’s positively-charged electron, or positron, in an experiment. It was the first time that a theoretical physicist had successfully predicted the existence of a previously unknown particle for purely mathematical reasons. Nowadays, theoretical physicists do this with gleeful abandon, and they are occasionally right. Dirac’s discovery utterly changed the rules of the game; the theoreticians no longer had to wait for experiments.
Opposite Electromagnetic particle shower. Particle tracks(moving from bottom to top) showing multiple electron-positron pairs created from the energy of a high-energy gamma ray photon.
The positron was also the first antimatter particle to be discovered. Physicists now understand that every particle has an antimatter equivalent; if a particle meets its antimatter twin, the two are annihilated. Thus Dirac’s formula led to a new and still unsolved problem: Why do we have more matter in the universe than antimatter? Why isn’t the universe empty?
Dirac’s equation also revealed that our universe has two fundamentally different kinds of quantum particle. Some particles have spin 0, ±1, ±2, etc., have vector wave functions, and are known as bosons. For example, photons fit into this category. Others, such as electrons, have spin ±1/2, ±3/2, etc., have quaternion-like (or “spinor”) wave functions,* and are known as fermions. All of the basic particles of ordinary matter—electrons, protons, and neutrons—are fermions.
Bosons like to congregate together; that is why lasers are possible. A laser beam is a collection of photons in the same quantum state. Fermions, on the other hand, stay aloof—you will never find two of them in the same quantum state. This is a good thing: it explains why atoms have electron orbitals. Because electrons can’t overlap, there is only room for two of them at the lowest energy level of an atom, eight at the next energy le
vel, and so on.
This pattern explains the periodic table and underlies all of chemistry. Imagine a universe without Dirac’s equation: it would be a universe with no matter as we know it, no chemical reactions, a universe with light and nothing else. A universe frozen at the first sentence of Genesis!
NOW LET’S STEP DOWN from the mountaintop and look at the more mundane applications of Dirac’s equation. They are legion. I have already mentioned lasers. Also, positrons are the fundamental ingredient in positron emission tomography (PET scans), used to study the activity of the brain. Electron spins are manipulated by magnetic fields in magnetic resonance imaging (MRI scans), a tool used to diagnose diseases without exposing patients to X-rays.
Finally, Dirac’s equation led quantum physicists to a new understanding of the vacuum, the ground state of the universe. They no longer see the vacuum as empty, but teeming with energy. Particles and their antiparticles can, and do, routinely pop into existence and pop right back out again. In fact, the whole concept of a “particle” is slightly outdated. To quantum physicists, the really fundamental concept is a quantum field. These fields, like electric fields, pervade all of space, and particles are their local manifestation. A particle is a fluctuation in the quantum field that may be just temporary or may be long-lasting.
Few equations in history have had more far-reaching implications. However, the man who discovered it, Paul Dirac, was notoriously taciturn and shy of publicity. If he spoke two words in a conversation, it meant that he was in a talkative mood. When he found out that he was going to receive the Nobel Prize in 1933, he initially wanted to decline it, until his friends persuaded him that declining the award would cause him to receive more publicity than accepting. Dirac largely escaped the public fascination and adulation that followed Newton and Einstein.
The Universe in Zero Words Page 14