In 1905 Albert discovered Relativity,
in 1906 he invented Rock and Roll
~Yahoo Serious

In the last week or two, I’ve gotten several excellent questions on special and general relativity. I’d like to devote this week’s post to presenting and answering those questions. For the sake of anonymity, I will call the people who asked the questions Ms. C and Mr. A. I hope you enjoy!

A Question on Special Relativity

The first question is by Ms. C, who asked:

I’ve read your article “The Speed of Light is Constant.” I’ve… got a question on the speed of light and aether…. Maxwell’s theory says the velocity of light is constant…[independently] of the observer’s speed. Now if light waves travel through the aether, wouldn’t their speed vary depending on my moving towards them or away from them? If yes, why did scientists believe in the aether-concept?

That’s a great question! You’re right; Maxwell’s theory says that the speed of light is constant, and if waves actually traveled through aether, the speed of a light wave would vary depending on your moving towards or away from it.

Despite this, scientists believed in the aether for about 40 years after Maxwell published his theory. I think that one reason for this is that scientists had a hard time imagining anything else. This isn’t their fault; although it seems easy in retrospect, the conceptual leap is a hard one.

Scientists first learned about waves by observing them in water and on a vibrating string, so it was initially very hard to even imagine a wave without some medium for it to propagate in. After all, a wave is the wiggling of something. If it travels through vacuum, what’s wiggling? The idea of wiggling fields is a bit more abstract than a wiggling aether.

Even Maxwell was convinced that the aether was true. When he originally unified electromagnetism, he published conditions on the aether to force it to conform to his equations. His description wasn’t complete, though, which is why we needed Einstein.

Because giving up the aether was conceptually difficult and Maxwell’s mechanical description of the aether didn’t quite fit, other scientists tried to modify Maxwell’s equations to fit with their intuition. There were actually quite a few attempts to resolve the paradox. You can read about them on Wikipedia:
http://en.wikipedia.org/wiki/History_of_special_relativity#Aether_models_and_Maxwell.27s_equations

In fact, right before Einstein discovered relativity, scientists got very close to a correct description of special relativity. Hendrik Lorentz actually came up with all the equations—length contraction, time dilation, etc.—that we needed for special relativity, but he still connected them to the aether.

Incidentally, Lorentz and Einstein were friends. At the beginning of Einstein’s career, Lorentz was a giant in the physics world; Einstein approached him as an admirer and the two eventually became quite close.

But Einstein went the other way. He trusted Maxwell’s mathematics more than he trusted his own intuition, and this is what allowed him to develop special relativity. He formulated his first postulate of special relativity–“[t]he laws of physics are the same in all inertial frames of reference”–specifically to forbid modifications to Maxwell’s equations.

This is actually pretty similar to a big problem we have in physics now. Both quantum mechanics and general relativity make correct predictions, but when we try to combine them, all of our attempts fail. People aren’t really sure what needs to be changed to treat gravity quantum mechanically. Is there something wrong with general relativity, or with quantum mechanics? We’re not sure. This is the problem of quantum gravity.

A Question on General Relativity

While Ms. C asked me about special relativity, Mr. A asked me about general relativity. He specifically asked me about my article “General Relativity Lets Us Take Shortcuts,” but I’ve also written an article on the geometry of spacetime and an article on the expanding universe. Here’s what Mr. A said:

As I understand it, Einstein’s General Theory of Relativity says that there is no gravitational force.  Although mass appears to attract mass, that is not actually happening.  Instead, mass curves spacetime so that the path of objects will be bent towards each other.In other words, when Einstein said that the gravitational force is fictitious, he did not mean that two objects will not move toward one another.  The part he thought was fictitious was the idea that they are moving together because their mass is pulling (or interacting with) each other.  The sole reason they move together is because their mass is curving spacetime, so that their paths converge.

For example, I throw a ball in the air and it comes back down to the ground.  Why did it come back down rather than keep going up into space?

1.)  Newton says it’s because the mass of the earth pulled the ball towards it.

2.)  Einstein says that’s wrong.  The mass of the earth did not pull on the ball.  Instead, the mass of the earth curved spacetime so that the path of the ball was bent back towards earth.  In other words, the mass of the earth directly affected the path of the ball, rather than the ball itself.

Is my premise accurate?… If that is true (there is no gravitational force), then how do you explain acceleration due to gravity?  Objects should accelerate only if a force is acted upon them; otherwise they should maintain a consistent velocity.

In the above example, I understand that the ball will fall to the ground because earth’s mass is curving its path toward the ground.  But if there’s no gravitational force acting upon the ball (i.e. the earth’s mass is not pulling on or directly interacting with the ball), shouldn’t it maintain its velocity on the way down?  What’s causing the ball to accelerate at 9.8 meters/second-squared?

Thanks for the question, Mr. A! This is a very sophisticated question with a very subtle answer, so I’ll go in steps. Your premise is essentially correct. However, there’s a big subtlety in the meaning of the word “path,” and explaining this subtlety will, I think, help answer your question.

But before I go into the meaning of a path in general relativity, I want to talk about acceleration.

Acceleration

The acceleration you’re talking about is a change in the speed of an object. If I’m driving along a straight highway and I press the gas pedal, my speed will increase. This is the type of acceleration you’re talking about. However, changing direction is also a type of acceleration, even if the overall speed doesn’t change. For example, if I drive a car on a circular race track at a constant speed of 60 mph (~90 kph), I’m constantly turning my car towards the center of the track. My direction is constantly changing; in this case, I’m always accelerating towards the center of the track.

To a physicist, these two types of acceleration are one and the same. The reason for this is because acceleration measures the change in velocity, which is both a speed and a direction of motion. Velocity and acceleration are what are called vectors. A vector is both a magnitude and a direction. You can imagine  velocity as an arrow, where the length of the arrow is the speed and the way the arrow points is the direction of motion.

This means that acceleration is also a vector, because it measures the difference between two velocity arrows. To find an acceleration, put the tails of the arrows for the old velocity and the new velocity together. The acceleration is the arrow that points from the tip of the old velocity to the tip of the new velocity. (A fiddly detail that I’m going to ignore: You have to scale the length of the acceleration vector by one over the amount of time it took to go from the initial velocity to the final velocity.)

Because acceleration is “just” an arrow, a change in speed with no change in direction (i.e., a linear acceleration) can cause the same acceleration vector as a change in direction with no change in speed.

Now, you might ask what happens when the initial velocity is zero. However, Einstein’s theory of special relativity tells us that there’s no such thing as “completely stopped.” I’m always moving with respect to someone else—and relative motion is the only thing we can actually define. This is why Newton’s first law of motion is:

An object at rest tends to stay at rest and an object in motion tends to stay in motion, unless acted on by an external force.

A modern interpretation of this law is that objects moving at a constant velocity tend to remain moving at that constant velocity. There is no such thing as “at rest.”

So this is one (very unsatisfactory) answer to your question. To Einstein, a change in direction and a change in speed are the same thing, so he didn’t need to explain both. This is one piece of the puzzle.

Let’s try and find a better answer.

 Curved Spacetime

In my article “General Relativity Lets us Take Shortcuts,” I explained that spacetime is curved and that the shortest path between two points is not always a straight line—in fact, this is what we mean by “curved.” Let’s explore this idea a little more. The curved surface we’re most familiar with is the Earth, so let’s see if we can’t get some feel for curvature by exploring how we move around on Earth.

Say you want to go from Narita, Japan to San Diego, U.S.A. What’s the shortest route? Naively, you’d look at a map and draw a straight line between the two cities. However, if you look at Japan Airline’s route map, this isn’t what you’ll see.

What’s going on? (Well, other than the effects of prevailing winds.) It’ll help if we look at the Earth as a sphere instead of as a plane. The straight line between the two cities goes through the Earth, so that’s a no-go. The naive path is just a straight line on a flat map, which in this case keeps our latitude more or less constant; this is doable, but not the best we can do. The best path is a path that goes a bit north.

What’s so special about this last path? Every path on the Earth must curve, because the sphere curves. However, there’s a portion of the curvature of the path that comes from the curvature of the Earth and there’s a portion that comes from the curvature of the path itself. The latter is called the geodesic curvature. A path that’s as straight as possible—i.e., whose only curvature is the curvature of the surface it’s on—is called a geodesic. This straightest possible path, which has no curvature of its own, will always also be the shortest possible path between two points.

The geodesics for planet Earth are the great circles. These are the circles with the same radius as the Earth; in other words, take the circle formed by the equator and rotate it to make it pass through any two points. A great circle will always cut the Earth into two hemispheres of equal size. However, they will no longer be the Northern and Southern Hemispheres we’re used to.

According to Einstein’s theory of general relativity, objects that are under no external force go in as straight a line as possible. This is the same as Newton’s first law of motion, but it takes the curvature of spacetime into account. How does spacetime become curved? Mass bends it.

Now, as Mr. A astutely pointed out, this explains why the path of an object curves when under the influence of gravity. If I throw a comet past the sun, the mass of the sun warps spacetime so that the shortest path past the sun bends around it in an arc. Alternatively, if I jump out of an airplane, spacetime warps such that the straightest path through it is down towards the Earth.

However, we still haven’t adequately answered Mr. A’s question. Why, if I jump out of the airplane, do I not only travel down towards Earth, but accelerate towards it at a rate of 9.8 m/s/s? What’s going on?

The secret is that Earth’s mass doesn’t just bend space—it bends time. The two can mix together and point in different strange directions with respect to each other. Furthermore, in relativity, a geodesic isn’t just the shortest path through space, but the shortest path through time.  You might ask what “shortest” even means if you’re travelling through time, but we can fall back on the correspondence between shortest paths and straightest paths.

In general relativity, a geodesic is the straightest possible path through space and time.

Once I jump out of the airplane, as I fall towards the Earth, time stretches out while the space between me and the Earth contracts. The straightest path I can follow through spacetime is the one where I accelerate at 9.8 m/s/s towards the Earth.

As strange and unintuitive as it is, this is the answer to Mr. A’s question. Mass bends our path through both space and time.

(There’s an important technical detail I’m glossing over. In special relativity, distances in time have a negative square length. This changes what the straightest possible path through spacetime is, which is an important part of why curved spacetime causes acceleration.)

A Big Thanks

A big thanks to both Miss C and Mr. A for emailing me these questions. I had a lot of fun answering them, and I hope you had fun reading my answers!

Questions? Comments? Hatemail?

As you can see, I love questions because it gives me something to write about. If you have any questions, comments, corrections, or even insults, please don’t hesitate to let me know! You can post in the comments, or you can email me at questions.thephysicsmill@gmail.com.