In 1907, while working at the patent office, Einstein had what he later called “the happiest thought of my life”: the equivalence principle. In simple terms, the principle says that the following two situations are locally indistinguishable:
Imagine a person standing inside an elevator on the ground. Now tell that person, “This cabin is floating freely in outer space.”
Since the person feels pressed against the floor, they would naturally conclude that the cabin must be accelerating upward.
From the viewpoint of the equivalence principle, we can picture the space around us as if it were falling toward Earth with an acceleration of 9.8 m/s2.
Let us examine gravity through the lens of the equivalence principle. Newton’s law gives the gravitational acceleration as
where g, G, M, and R are defined as follows.
| Gravitational acceleration g | 9.8 m/s2 |
| Gravitational constant G | 6.7×10−11 m3/s2 kg |
| Earth’s mass M | 6.0×1024 kg |
| Earth’s radius R | 6,400 km |
Multiplying both sides by 4πR2 gives
Interpreted through the equivalence principle, Earth may be pictured as a spherical surface of radius R that “draws in” space at a rate proportional to 4πR2×9.8 m3 per second squared. The right-hand side can likewise be read as Earth drawing in space at a rate proportional to 4π×6.7×10−11×M m3 per second squared. Because our universe has three spatial dimensions, this effect weakens in proportion to the square of the distance.
Einstein spent eight years developing the mathematics of the equivalence principle. He eventually realized that gravity could be described as the curvature of spacetime, an insight that led to the general theory of relativity in 1915.
The next figure illustrates the curvature of spacetime.
The blue line represents an object at rest when there is no gravity. The red line represents an object falling to the right in a gravitational field.
If we regard a moving object as being at rest in free space, its worldline itself can be understood as a representation of curved spacetime.
When explaining general relativity, people often use a curved rubber sheet as a model of space. Strictly speaking, however, it is not just space that is curved, but spacetime.
To see how matter and curvature are related, consider Einstein’s field equation:
Here, Gμν is a geometric quantity, while Tμν represents matter and energy. In this sense, matter tells spacetime how to curve. In his 1936 essay “Physics and Reality,” Einstein wrote:
[General Relativity] is sufficient — as far as we know — for the representation of experiences of celestial mechanics. But it is similar to a building, one wing of which is made of fine marble (left side of the equation), but the other wing of which is built of low-grade wood (right side of equation). The phenomenological representation of matter is, in fact, only a crude substitute for a representation which would do justice to all known properties of matter.
Einstein’s metaphor suggests that the right-hand side of the equation may also need a geometric description. In other words, matter itself might ultimately have to be understood geometrically.
What, then, is the true cause of gravity? Einstein’s equation relates matter to the curvature of spacetime, but it does not explain why matter bends spacetime in the first place. How does matter produce this curvature?
In 1870, mathematician William K. Clifford wrote the following to the Cambridge Philosophical Society (quoted in Introduction to Superstring Theory):
- That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.
- That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.
- That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or ethereal.
Perhaps matter itself is nothing more than curved space. If so, both sides of Einstein’s equation would be geometric, and the origin of gravity might finally become clear. For now, however, the ultimate cause of gravity remains unknown. Will you take up the challenge?
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