Figure 1. Is the Bering Strait exerting a mysterious attractive force on airplanes flying from Los Angeles to Taipei?
Einstein’s idea: Spacetime becomes curved
A mysterious force emanating from the Bering Strait
Imagine flying from Los Angeles to Taipei. Flipping idly through the back of an in-flight magazine (or more likely the flight map on the video these days), you might notice that the plane follows a curved path arcing toward the Bering Strait. Is the Bering Strait exerting a mysterious attractive force on the plane? See figure 1.
On your next trip you try another airline. This pilot follows exactly the same curved path. Don’t these pilots have any sense of originality? Why don’t they sometimes, just for the heck of it, swing south and fly over Hawaii, say? They seem to prefer flying over1 grim and unsuspecting Inuit hunters rather than cheerful Polynesian surfers.
Not only is the mysterious force attractive, it is universal, independent of the make of the airplane. Should you seek enlightenment from the guy sitting next to you? Dear reader, surely you are chuckling. You know perfectly well that the Mercator projection distorts the surface of the earth, and pilots follow scrupulously the shortest possible path between Los Angeles and Taipei. The answer to the universality of the mystery force is to be sought, not in the physics, but in the economics department.
We will come back to this story, but for now I digress.
Figure 1. Is the Bering Strait exerting a mysterious attractive force on airplanes flying from Los Angeles to Taipei?
A number divided by itself equals 1
Earlier, I had described Newton’s law of universal gravity, stating that the force F of gravitational attraction between a mass M and a mass m 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: F = GMm/R2.
In school we also learned Newton’s law of motion that the acceleration a of a body with mass m is equal to the force F exerted on the body divided by m: a = F/m.
Yes, it really is true, but tell that to a medieval peasant pushing a cart along a muddy road. He, and his educated contemporaries, would have regarded the claim that force produces acceleration as utterly loony. To all of them, and even to most of the proverbial guys and gals on our streets, Aristotle sounds much more plausible, claiming that force produces velocity. No force, no velocity.
The educated among us now understand that everyday life, alas, is dominated by friction, pain, and suffering. Aristotle appears to be right, and Newton wrong. But in fact Newton is right, and the venerable Greek, now banished from reputable physics departments everywhere, is wrong. The fundamental laws of physics do not know about friction, pain, and suffering.
So, the bottom line is: the acceleration of the moon due to the gravitational pull of the earth does not depend on the mass m of the moon at all. The force F is proportional to m, the acceleration a is given by the force F divided by m; ergo, the acceleration a does not depend on m.
This profound but elementary bit of math, that something divided by itself gives 1 (m/m = 1), indicates that all falling objects on the surface of the earth rush to the ground at the same rate. Again, we all learned in school that Galileo dropped cannonballs off the Leaning Tower of Pisa2 to see whether they would all hit the ground at the same time. Only a small fraction of school children now grown up, no doubt including my dear reader, remember why he did this. The rest of our fellow citizens would guess that Galileo was either loco or high.
Are inertial mass and gravitational mass really the same mass?
To Newton, mass corresponds to the amount of stuff.3 He quite naturally assumed that the mass m appearing in his law of gravity and the mass m appearing in his law of motion are one and the same.
But a hair-splitting lawyer, or a habitual reader of mysteries, would surely have detected a hidden assumption here. Are the blonde4 seen kissing the butler and the blonde caught leaving the house on the night of the murder really the same blonde? Are those two masses really the same mass?
To distinguish between the masses that appear in Newton’s law of gravity and in Newton’s law of motion, physicists called them the gravitational mass and the inertial mass, respectively. The former measures a couch potato’s obligation to listen to gravity, the latter his reluctance to get up and move. Conceptually, they are quite distinct and could very well not be equal.
Unlike the faculty in some other university departments, we in the physics department do not accept proofs by authority, not even a single-named giant in a likely apocryphal story. And thus the Hungarian Baron Loránd Eötvös de Vásárosnamény (1848–1919), instead of doing whatever barons did in the 19th century, devoted much of his life performing ever more precise experiments establishing the equality of the gravitational mass and the inertial mass. In our days, a series of experiments, known collectively as Eötvös experiments, have established the equality of the gravitational mass and the inertial mass to a fantastic degree of accuracy. In particular, an ingenious effort, led by my former colleague Eric Adelberger at the University of Washington, is fondly referred to as the Eöt-Wash experiment.5 Nerd humor in full force here!
Universality explained
That all objects fall at the same rate is known as the universality of gravity.
We now flash back to you sitting on the plane chuckling at the thought of your colleagues deducing that there must a mysterious force exerted by the Bering strait on airplanes.
But is it so laughably obvious? Consider the leading theoretical physicists before Einstein came along. They knew that all things fall at the same rate, be it an apple or a stone or a cannonball. To Einstein, that an apple and a stone would fall in exactly the same way in a gravitational field is no more amazing than different airlines, regardless of national or political affiliation, choosing exactly the same path getting from Los Angeles to Taipei. An apple or a stone traverses the same path in spacetime, just as a commercial flight follows the same path on the curved earth regardless of the airline.6 In hindsight, we might see an “obvious” connection, but hindsight7 is of course way too easy.
For 300 years, the universality of gravity8 has been whispering “curved spacetime” to us.
Finally, Einstein heard it.
We did not go looking for curved spacetime; curved spacetime came looking for us!
To Einstein, the equation
m = m
surely ranks as one of the two greatest equations in physics! The other is, of course,
c = c
No gravity, merely the curvature of spacetime
Just as there is no mysterious force emanating from the Bering Strait, one could say that there is no gravity, merely the curvature of spacetime. The gravity we observe is due to the curvature of spacetime. More accurately, gravity is equivalent to the curvature of spacetime: gravity and the curvature of spacetime are really the same thing.
To summarize and emphasize the point, Einstein says that spacetime is curved and that objects take the path of least distance in getting from one point to another in spacetime. Environment dictates motion. The curvature of spacetime tells the apple, the stone, and the cannonball to follow the same path from the top of the tower to the ground. The curvature of the earth tells the pilots to follow the same path from Los Angeles to Taipei.
This amazing revelation about the role of spacetime offers an elegantly simple explanation of the universality of gravity.
Gravity curves spacetime. That’s it.
Spacetime is curved and gravity’s job is done. It’s now up to every particle in the universe to follow the best path in this curved environment. This explains why gravity acts indiscriminately on every particle in exactly the same way. Next time you take a nasty fall, whether on the ski slope or in the bathtub, just think, every particle in your body is merely trying to get the best deal for itself. Best deal? To be explained in chapter 9.
Curved spacetime
“Space tells matter how to move and matter tells space how to curve.” This memorable summary9 of Einstein gravity, due to John Wheeler, my first mentor10 in theoretical physics (as was mentioned earlier), has been widely publicized.* More accurately, for “space” we should say “spacetime.”
If I were an intelligent layperson reading popular physics books, I would have been exceedingly frustrated by the term “curved spacetime.” These days, even the mass media bandy the term “curved spacetime” about with some abandon. But what exactly does it mean to say that “spacetime is curved?”
I have addressed the appendix to readers like me. For those readers who do not wish to tangle with some math, no matter how slight, we can do fine proceeding by analogy.
When we think about curved surfaces, such as the surface of a balloon, we see it as living in an ambient 3-dimensional flat space, the plain old Euclidean space we were born into and will die in. In math speak, the curved 2-dimensional surface is said to be embedded in a higher dimensional flat space. But as indicated in the appendix, we can perfectly well conceive of, and describe, a curved space or spacetime without having to embed it in a higher dimensional space or spacetime.
The metric of spacetime
Let us go back to the water wave described in chapter 4. Recall that we specify the surface of a pond by the height of the water measured from the bottom. At time t, and at the location specified by (x, y), call the height g (t, x, y), a function that depends on time t and space coordinates x and y. Without any breeze whatsoever, the surface of the pond is flat and thus g (t, x, y) = 1. But in general, g (t, x, y) varies in time and space in some complicated way according to an equation written down in the 19th century.
However, when the waves are gentle and relaxed, that is, in the linear regime,* we can write g (t, x, y) = 1 + h(t, x, y) and treat h as small compared to 1. Then the equation we have to deal with simplifies.
As was already mentioned in part I, the situation with Einstein gravity is closely analogous to the story of water waves. Einstein’s field equation governing the curvature of spacetime is essentially impossible to solve in general, but it simplifies enormously for gravity waves in the linear regime, so that most physics undergrads should be able to solve the corresponding equation.
However, several technical, rather than conceptual, complications manage to befuddle many physics undergrads. But in a popular book, rather than a textbook,11 we can readily breeze by these complications.
First, the simplest complication: in Einstein gravity, the analog of the quantity g becomes a function (or more strictly speaking, field) g (t, x, y, z) of time and three spatial coordinates, namely Decartes’s x, y, z of the 3-dimensional space we live in.
Second, to describe the curvature of spacetime, we need ten* such functions instead of one. But unless you want to get an advanced degree in physics, you need not be concerned.
Third, in the case of water waves, when the surface of the pond is flat, g = 1. Similarly, when spacetime is flat (that is, in the absence of gravity waves), these ten fields g (t, x, y, z) are constant and equal to a simple number. The slight complication is that, of the ten, three are equal to 1, one is equal to − 1, and the rest equal to 0. (Aren’t physics and math fun?)
These ten fields g (t, x, y, z), known as the metric of spacetime, determine the distance between two neighboring points in spacetime. Given the metric, we can deduce† the curvature of spacetime.‡
I can give you a vague sense of how this works using an everyday example. Given an airline table of distances, you could deduce that the world is curved without ever going outside. If I tell you the three distances between Paris, Berlin, and Barcelona, you could draw a triangle on a flat piece of paper with the three cities at the vertices. But now if I also give you the distances between Rome and each of these three cities, you would find that you could not extend the triangle to a planar quadrangle. So the distances between four points suffice to prove that the world is not flat.
But the metric tells you the distances between an infinite number of points. The reason is that once we know the distance between neighboring points, we can add up these tiny distances to find the distance between any two points.
On a world map in Mercator projection,* Greenland looks bigger than China, but you know that Denmark, to which Greenland belongs, does not rank in the top ten countries by area (figure 2). From this fact alone you could deduce that the world is curved. Once the metric tells you about distance, it also tells you about area.
* Frankly, I do not find this formulation so exceptional. Already in Newtonian gravity, the gravitational field tells matter how to move, and matter tells the gravitational field how to behave. And in electromagnetism, the electromagnetic field tells charges how to move, and charges tell the electromagnetic field what to do.
* This bit of jargon was introduced in chapter 4.
* The number ten will be explained in the appendix.
† That is, the mathematicians Gauss and Riemann figured out in the 19th century how to calculate the curvature given the metric.
‡ As I’ve already said, the reader who wants more can find more in the appendix to this book.
* After Gerardus Mercator, namely, Jerry the Merchant. In mapping the round sphere to a flat piece of paper, Mercator preserves the angles between straight lines but not the distance between points. For those who are lost, knowing the direction to your destination is more important than knowing how far you are from your destination.
