On Gravity

A Brief Tour of a Weighty Subject

Anthony Zee

• Publisher: Princeton University Press
• Year: 2018
• ISBN:
  9781400890309
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

A brief introduction to gravity through Einstein's general theory of relativity Of the four fundamental forces of nature, gravity might be the least understood and yet the one with which we are most intimate. From themonths each of us spent suspended in the wombanticipating birth to the moments when we wait for sleep to transport us to other realities, we are always aware of gravity. In On Gravity, physicist A. Zee combines profound depth with incisive accessibility to take us on an original and compelling tour of Einstein's general theory of relativity.Inspired by Einstein's audacious suggestion that spacetime could ripple, Zee begins with the stunning discovery of gravity waves. He goes on to explain howgravitycan be understood in comparison to other classical field theories, presents the idea of curved spacetime and the action principle, and explores cutting-edge topics, including black holes and Hawking radiation.Zee travels as far as the theory reaches, leaving us with tantalizing hints of the utterly unknown, from the intransigence of quantum gravity to the mysteries of dark matter and energy.Concise and precise, and infused with Zee's signature warmth and freshness of style, On Gravity opens a unique pathway to comprehendingrelativityand gaining deep insight into gravity, spacetime, and the workings of the universe.

Part I

1

A friendly contest between the four interactions

Matter and the forces that move it

To tell the story of gravity waves, let us first go for a quick tour of the universe. Matter consists of molecules, and molecules are built out of atoms. An atom consists of electrons whirling around a nucleus, which in turn consists of protons and neutrons, collectively known as nucleons. The nucleons are made of quarks. That’s what we know.*

The universe also contains dark matter and dark energy. (More in chapter 18.) Indeed, by mass, the composition of the universe is 27% dark matter, 68% dark energy, and only 5% ordinary matter. To first approximation, the universe may be regarded as one epic cosmic struggle between dark matter and dark energy.¹ The matter we know and love and of which we are made of hardly matters. Unhappily, at present we know little about the dark side.

We know of four fundamental forces between these particles. When particles come into the vicinity of each other, they interact,^† that is, influence each other. Here is a handy summary of the four forces, known as gravity, electromagnetism, the strong interaction, and the weak interaction.

G: Gravity keeps you from flying up* to bang your head on the ceiling.

E: Electromagnetism prevents you from falling through the floor and dropping in on your neighbors if you live in an apartment.^†

S: The strong interaction causes the sun to provide us light and energy free of charge.

W: The weak interaction stops the sun from blowing up in your face.

I don’t quite remember, but I would suppose that, due to buoyancy,^‡ we were not aware of gravity while in our mothers’ wombs. But as soon as you entered the world, you knew about gravity, especially if the obstetrician grabbed you by the ankles and hanged you upside down. Then that quick slap on your bottom caused you to cry out and to open your eyes, thus discovering electromagnetism.

Only four forces!

The world appears to be full of mysterious forces and interactions. Only four?

As you toddled, you banged your head against a hard object. What is the theory behind that? Well, the theory of solids can get pretty complicated, given the large variety of solids. But a simple cartoon picture suffices here: the nuclei of the atoms composing the solid are locked in a regular lattice, while the electrons cruise between them as a quantum cloud. A collective society in which all individuality is lost! The atoms no longer exist as separate entities. The arrangement is highly favorable energetically; that is jargon for saying that enormous energy is required to disturb that arrangement. Revolution is costly. It takes quite a tough guy to crack a rock into halves.

So, the myriad interactions we witness in the world, such as solid banging on solid, can all be reduced to electromagnetism. What we see in everyday life is by and large due to some residual effect of the electromagnetic force. Since common everyday objects are all electrically neutral, consisting of equal numbers of protons and electrons, the electromagnetic force between these objects almost all cancels out. Even the steel blade of a jackhammer smashing into rock is but a pale shadow of the real strength of the electromagnetic force.

Just about the only time the true fury of electromagnetism shakes us is when thunder and lightning fill the sky. While we modern dudes have totally enslaved electromagnetism, all ancient people attribute its occasional bursts of temper to the gods.²

When you first shook off the ooze, you might have thought that there must be thousands, if not millions, of forces in the world. Thus, to be able to state that there are only four fundamental forces is totally awesome, a feat summarizing centuries of painstaking investigations. For example, realizing that light was due to electromagnetism stands as a towering achievement.

The universe as a finely choreographed dance

While the proverbial guy and gal in the street are plenty acquainted with gravity and electromagnetism, they have no personal experience with the strong and the weak interactions. But in fact, the physical universe is a finely choreographed dance starring all four interactions.

Consider a typical star, starting out life as a gas of protons and electrons. Gravity gradually kneads this nebulous mass into a spherical blob in which the strong and the electromagnetic forces stage a mighty contest.

The electric force causes like charges to repel each other. Thus, the protons are kept apart from each other by their mutual electric repulsion. In contrast, the strong force, also known as nuclear attraction, between the protons tries to bring them together. In this struggle, the electric force has a slight edge, a fact of prime importance to us. Were the nuclear attraction between protons a tiny bit stronger, two protons could get stuck together, thus releasing energy. Nuclear reactions would then occur very rapidly, burning out the nuclear fuel of stars in a short time, thereby making steady stellar evolution, let alone civilization, impossible.

In fact, the nuclear force is barely strong enough to glue a proton and a neutron together, but not strong enough to glue two protons together. Roughly speaking, before a proton can interact with another proton, it first has to transform itself into a neutron. The weak interaction has to intervene to cause this transformation. Processes affected by the weak interaction occur extremely slowly, as the term “weak” suggests. As a result, nuclear burning in a typical star like the sun occurs at a stately pace, bathing us in a steady, warm glow.

Range versus strength

The reason that the proverbial guy and gal in the street do not feel the strong and the weak interactions is because these two interactions are short ranged. The strong attraction between two protons abruptly falls to zero as soon as they move away from each other. The weak interaction operates over an even shorter range. Thus, the strong and weak interactions do not support propagating waves. In this book, we won’t talk about these two short range interactions much.

A boxer with short arms but a strong punch versus a boxer with long arms but a weak punch.

From Fearful Symmetry: The Search for Beauty in Modern Physics by A. Zee. Copyright ©1986 by A. Zee. Princeton University Press.

In contrast, the gravity force between two masses and the electric force between two charges both fall off with the separation R between the two objects like 1/R², the inverse square law celebrated in song and dance. More on this in chapter 2. Gravity and electromagnetism are known as long ranged and thus can and do support propagating waves.

For R large, these forces still go to zero, but slowly enough that we can feel the tug of the sun, literally an astronomical distance away. For that matter, our entire galaxy, the Milky Way, is falling toward our neighbor, the Andromeda galaxy.

Thus, in the contest among the four interactions, brute strength is not the only thing that counts: many phenomena depend on an interplay between range and strength. A case in point is fusion versus fission in nuclear physics. When two small nuclei get together, each consisting of a few protons and some neutrons, the strong attraction easily overwhelms the electric repulsion, and they want to fuse. In contrast, in a large atomic nucleus (famously, the uranium nucleus), the electric repulsion wins over the strong attraction. Each proton only feels the strong attraction of the protons or neutrons right next to it, but each proton feels the electric repulsion from all the other protons in the nucleus. The nucleus wants to split into two smaller pieces, accompanied by the release of energy.

* Whether quarks and electrons are tiny bitty strings is an intriguing, but at the moment purely speculative, possibility.

† “Interact” is a technical word in physics, just like “energy,” “momentum,” and “mass.”

* You know how fast the earth is spinning to cover about 24,000 miles in 24 hours. Anybody who has studied some physics could calculate what the centrifugal acceleration would be.

† Plus a lot of other good deeds. Electromagnetism holds atoms together, governs the propagation of light and radio waves, causes chemical reactions, and last but not least, stops us from walking through walls.

‡ A force driven in fact by gravity, as the fluid around you fought for a better deal by getting lower.

2

Gravity is absurdly weak

Gravity and the electric force

Gravity is absurdly weak compared to the electromagnetic force.

How do we compare the relative strength between two forces at the fundamental level? First, a reminder of some basic facts.

We learned about Newton (1642–1726/27)¹ and his law of universal gravity in school. It states that the force F of gravitational attraction between a mass M (say, the earth) and a mass m (say, the moon) is equal to a constant G (known as Newton’s gravitational constant) times the product of the two masses (namely, Mm) divided by the square of the distance R separating them. Or, in a more concise language, F = GMm/R².

We also learned about Coulomb’s law. It sta...