Of Particular Significance

# Chapter 4, Endnote 6

Astronauts in a space station seem to float, as though they are weightless. Are they truly weightless? Or are they only apparently weightless?

The same issues arise for people in a freely falling elevator, accelerating downward with ever greater speed. They will feel weightless, too. But are they?

Newton would have said they are apparently weightless, subject to gravity but all falling together along with their vehicle. A naive (but instructive!) reading of Einstein might lead us to say that they are truly weightless… that the gravity that Newton claims is present is a pure illusion, a fictitious force. But a precise Einsteinian would say they are almost but not quite weightless — and the lack of perfect weightlessness is a clue, a smoking gun in fact, that they are indeed subject to gravity.

### NEWTON’S VIEWPOINT

The Newtonian view is that a space station, elevator, and any person within them is subject to the pull of a gravitational force due to the Earth, and that this force is proportional both to the mass M of the Earth and the mass m of the object being pulled. Meanwhile, Newton’s second law of motion states that the acceleration due to a force is inversely proportional to the mass of the object being pulled. Since the force of gravity is proportional to the object’s mass while the acceleration caused by the force is inversely proportional to the mass, the effect of Earth’s gravity on an object’s motion is independent of the object’s mass(This simple view holds as long as the object’s mass is small enough that it in turn does not affect the Earth’s motion! Otherwise the discussion becomes somewhat more complicated.)

As a result of this, a freely falling elevator and the people inside it accelerate together, traveling always at exactly the same speed. Since all their motions are the same, nothing causes the people to move either toward the elevator’s ceiling or toward its floor. And so, if these people are in the middle of the elevator, perhaps even upside-down or sideways from our perspective, nothing seems to pull them in any direction. They will sense themselves as floating inside the elevator, as though there were no gravity. They will feel weightless, even though they and their elevator are falling more and more rapidly thanks to the pull of gravity — thanks, in other words, to their weight.

The same is true of a person orbiting the Earth inside a space station; indeed, an orbit is just a special way of falling forever, where one moves around the Earth fast enough that falling downward combined with moving forward leads to circular motion. (See the book’s Figure 9.) Just as do people in a falling elevator, astronauts in an Earth-orbiting space station move in exactly the same way as does their spacecraft, with nothing drawing them toward any of the craft’s walls. That’s why they will feel as though they are motionless and floating — as though they are weightless.

From our perspective, watching our friends in free fall or in orbit, we know that they are subject to the Earth’s gravity. The elevator would not be falling, nor the spacecraft orbiting, were there no gravity from the Earth; and were the people inside these conveyances not subject to gravity, they would not travel in concert with their vehicles and would not feel weightless. Their experience of apparent weightlessness is caused by the universality of gravity, not by the absence of gravity, according to Newton.

### EINSTEIN’S VIEWPOINT — THE EQUIVALENCE PRINCIPLE

Now, what does Einstein say about all this? Einstein makes a claim known as the principle of equivalence. Its simplest form is this: if the effects of gravity are constant across all of space, then one cannot say that there is any gravity at all, because there is no physical distinction between a spatially-uniform gravitational force and a special form of pure motion: constant acceleration. This has led some people to say that gravity is an illusion, a fictitious force.

The classic example of a fictitious force is what is called centrifugal force — a force pushing away from a central point — such as the force that seems to push you toward the right side of a car when it turns to the left, and vice versa. The fact that centrifugal forces can be viewed as fictitious is made clear by the fact that there always exists a perspective from which the centrifugal force is exactly zero. As your car drives past me and turns left, I see no sign of your apparent centrifugal force. Instead, what I see is the car experiencing a centripetal force — a force toward a central point — created by the friction between the road and the tires; this is what makes it turn left. You, not feeling this force, tend to go straight, and thus you move to the right relative to the car. From your perspective, the car moving left underneath you due to a leftward force acting on the car feels as though you are subject to a rightward force; but this is an illusion.

Yes, the Newtonian force of gravity is somewhat illusory in Einstein’s view. As Einstein realized, Newton’s force is relative — that is, it is perspective-dependent, much like centrifugal force. That means that it doesn’t make sense to specify its strength and direction in the way that Newton would have recommended; different observers will view it differently.

But still, gravity is not like centrifugal force; it’s not an illusion. There are aspects of gravity that are intrinsic and not relative — they are independent of one’s perspective — and because of this, gravity is not to be viewed as fictitious. It’s certainly not a Newtonian force. But it is most definitely a force in the modern physics sense. (And it’s not alone in being partly ficticious: even electric and magnetic forces have aspects that are relative and others that are intrinsic. For instance, from your perspective, a stationary electrically charged particle is surrounded by nonzero electric field but has zero magnetic field; however, if I travel rapidly past you, I will claim that the magnetic field around the particle is nonzero. Both of us are right.)

A separate argument sometimes made in this context is that in Einstein’s view, gravity is a manifestation of the curvature of space and time, and so, it is claimed, gravity isn’t a real force. But this has no bearing on the question of whether the gravity is fictitious. Space-time curvature causes effects that we can choose to interpret as forces, and there are aspects of those forces which are not illusory.

Einstein’s view does imply that for persons inside an isolated bubble (such as a closed elevator or a space station far from other objects), the force of gravity cannot be measured unambiguously. Here’s another example. Suppose, for instance, there are persons in an elevator who feel just as they normally would in a stationary room on Earth — as though they were held by uniform gravity to the elevator’s floor, as in Fig. 2. No experiment can reveal whether they are indeed held by a constant gravitational force to the floor (as we are, almost exactly, at the Earth’s surface) or whether the elevator is subject to no gravity at all, and instead is undergoing a constant acceleration due to a force that pushes the elevator but not the persons inside. The floor, in contact with the persons’ shoes, is pushing them, causing them to accelerate upward with the elevator, and the experience of that upward acceleration is indistinguishable from the experience of constant downward gravitational pull.

Similarly, if they feel weightless, like the person in Fig. 1, some of them might imagine they are gently floating in space, far from any object that could cause gravity. But others might worry that perhaps they are falling down an elevator shaft just outside a giant star, moments from certain death. If they try to think of experiments to determine which of them is right, they will fail, because of the principle of equivalence: no experiment of any type can distinguish motion caused by a uniform gravitational force from perfect weightlessness. This is the central pillar of general relativity, Einstein’s giant step beyond Galileo’s relativity. But that’s a subject for another book.

It might seem, then, that whether gravity is present or absent is not subject to experimental test, and that its presence is wholly a matter of perspective. But this is only true if the gravitational force is perfectly uniform across the bubble. Certain variations in gravity across the bubble can be measured unambiguously.

[Caution: here I am oversimplifying; it’s not only uniform gravitational fields that are fictitious. For instance, any choice of non-Cartesian coordinates on flat space-time , such as Rindler coordinates, creates such forces. To fully understand when and why certain combinations of gravity and acceleration are fictitious, while others are not, requires more of the math and concepts of general relativity.]

Gravity created by the presence of real-world objects is never perfectly uniform. The strength of Earth’s gravity varies with one’s distance from the Earth’s center, and its direction varies with one’s location. Let’s see now that these gravitational forces are not fictitious.

### EINSTEIN’S VIEWPOINT — TIDAL FORCES

The essential point is that gravity around any finite object is not uniform. It varies, and that makes all the difference, as illustrated in Fig. 3.

Let’s consider our friends in the elevator who want to know if they are truly weightless in deep space or are falling in an airless shaft toward the Earth’s center. If they are armed with precise instruments (or if the elevator is unusually tall) they will notice that there are in fact gentle forces in the elevator — that objects tend to fall toward the elevator’s floor and to rise toward its ceiling. This is because gravity’s pull is slightly stronger near the floor, and slightly weaker near the ceiling, than it is on average across the elevator. The effect is straightforward to measure with precise instruments, even though it is far too small for a human to feel. [It would not be too small if the observers were falling just above a neutron star, however.] One consequence is that any object that extends vertically along the elevator tends to be slightly stretched. Not only that, this stretching effect will change over time. (If the elevator is falling from space toward the Earth’s surface, the tidal effect will grow. If the elevator shaft is already underground, as in a mine, it will shrink.)

These subtle forces are tidal forces — aspects of gravity that, unlike gravity’s overall strength in a small isolated bubble, are not matters of perspective. Tidal effects are intrinsic, not relative; there’s nothing illusory about them. They actually stretch and squeeze physical objects, and you can’t make them disappear simply by changing your viewpoint.

This is the key insight: by measuring the presence of tidal forces, the inhabitants of the elevator can confirm that they are not weightless. They will not be able to determine exactly what is happening to their elevator purely from measurements of the tidal forces, but they can use them to rule out various possibilities, including the hypothesis that they are drifting in deep space, unaffected by any other objects.

### THE GLOBAL PERSPECTIVE

In this discussion of whether gravity is an illusion, we have focused on what is observable by an observer with a narrow perspective inside an isolated bubble. Why, however, are we privileging the inhabitants of an elevator or space station? Yes, from their point of view, they are almost weightless; but we, viewing from outside, have access to vastly more information, and it makes no sense to pretend we don’t have it. For instance, we might be able to observe thousands of falling elevators and spacecraft all around the Earth, each with its floating space crew. From their motions and from the real tidal forces they experience, we can map out the area around the Earth, and see that it is accompanied by tidal forces and motions completely consistent with a spherical gravitational effect centered on the Earth itself, one that causes objects to accelerate toward the Earth’s center. Whether we describe this effect as a Newtonian-like force is a matter of perspective, but the tendency of objects located anywhere nearby to approach the Earth and crash into its surface clearly is not. (A collision cannot be made to un-happen simply by changing one’s point of view!)

So it’s misleading to say that gravity is an illusion and that the space station astronauts are truly weightless. The space station astronauts can measure the tiny tidal forces around their station, and confirm they are not exactly weightless and are subject to some source of gravity. Meanwhile, their space station is one of many satellites of the Earth, and we, watching them all, can confirm that they are subject to Earth’s gravitational effects (even if we view those effects as due to spacetime curvature and avoid using the word “force” altogether.) Though the strength of gravity’s pull as Newton defined it is perspective-dependent, and not reliable, there is nothing debatable about the statement that it’s Earth’s gravity that keeps the space station and its inhabitants in orbit. (For a related discussion, see herehere and here.)

Additional discussion of tidal forces will be found in Chapter 4, Note 12.

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