###### By Dan Hooper, Ph.D., University of Chicago

## The special theory of relativity, brilliant as it was, still had some holes to account for after its initial conception. Einstein’s equivalence principle would help to address these issues, though the equations would take years to refine.

### The Special Theory of Relativity: Limitations

Einstein’s theory of relativity—now known as the special theory of relativity—describes how lengths in space and durations of time are different to observers moving at different speeds, in different frames of reference.

Although special relativity predicts many observed phenomena correctly, this theory is also incomplete. Even Einstein himself was aware that it was incomplete from early on.

Firstly, special relativity can only be applied to objects that are moving at a constant rate of speed. In this sense, special relativity is like a theory that describes how a car moves and behaves with its cruise control on but knows nothing about the brake pedal or the accelerator.

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Secondly, in the Newtonian theory of gravity, the accepted theory of the time, gravity’s attraction works instantaneously, pulling bodies together across great distances in space without any time delay. According to special relativity, nothing can move faster than the speed of light.

This made it difficult to reconcile special relativity with this aspect of Newtonian gravity. In at least two ways, Einstein’s theory of special relativity left us with important and unanswered questions.

Shortly after publishing his special theory of relativity, Einstein began to work toward creating an even more complete and far-reaching theory of space and time. Although it took him another decade, Einstein eventually came up with an expanded and completely general form of his theory.

The general theory of relativity was not only a theory of space and time, but also provided us with a deeper, more powerful way of thinking about the force of gravity.

### Introducing the Equivalence Principle

In 1907, Einstein had his first important conceptual breakthrough that placed him on the road to general relativity. This occurred a few years after special relativity and his other breakthrough papers from 1905.

Thinking about how he might be able to incorporate acceleration and gravity into his theory, he came up with something we now call the equivalence principle.

To understand this concept, imagine that you’re in an impenetrable chamber—you can’t hear, see or otherwise know anything about what’s going on outside of the chamber. Toward one side of the chamber, you feel a force that feels just like gravity does.

It pulls you toward one side of the chamber, and it allows you to walk normally along what feels like the bottom of the chamber. But is this genuinely the force of gravity?

Instead, what feels like gravity to you might be the consequence of the chamber being accelerated. When you’re in an elevator that’s speeding up or accelerating, you feel a downward force that makes you feel slightly heavier than normal. When the elevator is slowing down, you feel an upward force, making you slightly lighter.

The fact is that the force of gravity feels the same as the effects of acceleration. To someone sealed in the chamber, there is no way to know whether the force that they are experiencing is in fact gravity, or is instead the consequence of the chamber being accelerated.

This is the essence of Einstein’s equivalence principle. Although he didn’t know yet where it would lead him, this insight made Einstein begin to speculate that acceleration and gravity might be very deeply interconnected.

Learn more about Einstein’s special theory of relativity

### Relating Mass, Gravity, and Acceleration

To better appreciate the nature of the equivalence principle, consider what we mean when we use the word “mass”. In Newtonian physics, there are two very different kinds of quantities that we sometimes call “mass”.

The first of these is the kind of mass that resists acceleration. We call this inertial mass. Something with a lot of inertial mass—like a boulder, for example—requires a lot more force to move than something with much less inertial mass—like a baseball.

The second kind of mass is what gravity acts upon. We call this kind of mass gravitational mass. The weird and surprising thing is that the inertial mass of an object seems to be exactly equal to its gravitational mass.

As far as we know, there are no objects in our universe with more inertial mass than gravitational mass or vice versa. For some reason—unknown before Einstein—the inertial mass and gravitational mass of an object were always the same.

But Einstein’s equivalence principle provided us with an insight as to why this was the case. He began to think that the force of gravity was just acceleration in some sense. If this was the case, then it might not be surprising at all that gravitational mass was just the same thing as inertial mass.

Learn more about Einstein and gravitational waves

### Gravity and Sunlight

Well before Einstein constructed his theory of general relativity, he recognized a particularly important consequence of the equivalence principle—beams of light should be subtly deflected or bent by the force of gravity. In 1911, he published an article that pointed this out.

He entitled this article “On the Influence of Gravity on the Propagation of Light”, and in it, Einstein presented a calculation showing that a ray of light passing by the Sun should be deflected by about 0.83 arcseconds, or about one four-thousandth of a degree. A very subtle effect, but one that could be tested, at least in principle.

But under normal circumstances, any light that was deflected by the Sun would be lost in the much brighter sea of ordinary sunlight. To see or detect the deflected beam of light as it skims past the Sun, the light of the Sun would have to be blocked out.

For such a measurement to succeed, it would have to be made under the conditions of a nearly perfect solar eclipse. The next solar eclipse was predicted to take place three years later in 1914. At that time, Einstein hoped that his prediction—and the equivalence principle along with it—would be proven correct.

Einstein spent the years leading up to the scheduled eclipse considering some of the conceptual questions that were raised by the possibility of the gravitational deflection of light. In many applications, beams of light had long been used as the very definition of a “straight line”.

If the Sun’s gravity could bend the trajectory of a ray of light, then—at least in some sense—gravity could change the geometry of space. With this insight, Einstein began to recognize the deep connection that exists between what we call gravity and the geometry of space and time.

But even Einstein was not yet in any position to truly understand this connection. To build the theory he was beginning to imagine, Einstein would have to dig much deeper into the mathematics of geometry, deeper than any physicist had ever gone before.

Learn more about how Einstein proved that light is made up of discrete quanta

### Euclidean Geometry: The Standard for Math

In high school, you probably took a geometry class, where you were almost certainly taught a system known as Euclidean geometry.

Until Einstein came along, physics was entirely based on Euclidean geometry. To almost everyone at the time, Euclidean geometry was seen as the only reasonable way to think about space.

Euclidean geometry is named after the ancient Greek philosopher and mathematician Euclid. Everything about it can be derived from five basic rules, sometimes called axioms or postulates. These postulates seem very self-evident.

For example, one of Euclid’s postulates states that “any two points in space can be connected by a straight line”. And another says that “all right angles are equal to each other”.

But Euclid’s fifth postulate turns out to be on less solid footing. The fifth postulate states that “for any straight line there is exactly one straight line that is parallel to it that passes through any given point in space”.

Among other things, this last postulate can be used to show that two parallel lines will never meet or cross one another. In your high school geometry class, you were probably taught this postulate as an indisputable fact. It seems obvious.

Throughout most of history, Euclid’s postulates were treated as self-evident and indisputable. But in the first half of the nineteenth century, a few mathematicians started to think about systems of geometry that broke one or more of these postulates.

Learn more about Einstein’s deterministic version of quantum theory

### Do Geometric Rules Always Apply to Reality?

Many mathematicians had managed to develop self-consistent geometrical frameworks that do not adhere to the fifth postulate’s position on parallel lines. In these new non-Euclidean geometries, two parallel lines do not necessarily remain parallel.

Instead, two straight lines that are parallel to each other at one point in space can come together or diverge from one another as you follow them along their paths. In these geometrical systems, it can be shown that the three angles of a triangle don’t always have to add up to 180 degrees—they can add up to a larger or a smaller number.

The ratio of a circle’s circumference to its diameter doesn’t have to be equal to the number pi. Within these non-Euclidean systems, much of what you learned in high school geometry turns out not to be true.

Just because a mathematician can write down a weird geometrical system, doesn’t mean that it’s real in any physical sense. Mathematics is certainly useful to physicists, but not all mathematical possibilities are realized in nature.

What these 19th-century mathematicians had done was to prove that logic and reason alone don’t force us to accept Euclidean geometry; there are other self-consistent possibilities.

Learn more about the phenomenon of entanglement

Intrigued by these strange new systems of geometry, a handful of mathematicians and physicists began to consider whether they might have anything to do with our physical world. But despite a few intermittent shows of interest, most physicists didn’t take these exotic geometries seriously.

That was until Einstein placed them at the very heart of the general theory of relativity.

### Common Questions About Einstein’s Equivalence Principle

**Q: What exactly is Einstein’s Weak Principle of Equivalence?**

The Weak Principle of Equivalence concerns the laws of motion of bodies in free fall and states they are the same as in an unaccelerated reference frame.

**Q: What is the importance to general relativity of Einstein’s Equivalence Principle?**

Einstein’s Equivalence Principle is crucial to Einstein’s theory of general relativity in that it states that mass is the same whether inertial or gravitational, and so these types of movement are not altered by mass.

**Q: Why can nothing move faster than the speed of light?**

Einstein’s statement that the speed of light is the speed limit in the universe hinges on his discovery that particles gain mass as they accelerate and thus would require infinite amounts of energy to accelerate to the speed of photons which have no mass.

**Q: What exactly is Einstein’s General Relativity?**

Einstein’s Principle of General Relativity essentially states that spacetime is curved and the laws of physics remain the same in any inertial frame of reference but can change in non-inertial frames.