Galileo Galilei

Isaac Newton

Gottfried Leibniz

Albert Einstein

Space and time separations are not the same in different reference frames.

speed of light,

c = 299,792,458 m/s

*The speed of light, as stated above, is only the speed of light in a vacuum (or through "empty" space). Light slows down when traveling through transparent materials, like water and glass.

Let's take a break from the details of modeling moving objects to examine motion more broadly. The goal here is to establish a background within which our kinematic equations (our four models of constant-acceleration motion) play out.

Imagine that you are on a train, headed north at 17.8 m/s, with a soccer ball resting against your feet. (I know it is difficult for an American to imagine riding a train; pretend you are traveling in France.)

A kid sitting across the aisle asks you, "Is that your ball?" You answer, "Yes." Then he strangely asks, "Is your ball moving?" You hesitate to answer, but you admit, "No, it's obviously resting against my foot." He grins, "It IS moving! We're on a train!" And he runs away laughing.

The kid may be annoying, but he has brought up a very deep idea. What does it mean to be moving?

From one point of view (specifically, yours and the kid's, before he leaves his seat) the soccer ball is at rest. From the point of view of someone standing outside the train, the ball is certainly not at rest, and neither are you. This is what the kid was alluding to. Okay, so is the ball actually moving at 17.8 m/s to the north? Is that its "real" or actual velocity?

"No!" shouts Marvin the Martian, spying on you from his home planet, using a very powerful telescope. The ball (and you) are stuck on the Earth, hurtling through space at thousands of meters per second, at least from the Martian's point of view.

So what IS the ball's true or actual velocity? Well, it doesn't have one! Its measured velocity depends on one's perspective. And there is no correct velocity or correct perspective. There are only convenient and inconvenient perspectives.

Realize that when you specify the velocity or speed of something, it is always with reference to some other object, even if that object goes unnamed. When you say that a car is driving at 60 mph, you mean that the car is going 60 mph faster than the patch of Earth beneath the car. A car at rest is only at rest relative to the ground beneath it, i.e. the distance between the car and the Earth beneath it is not changing with time.

We could say of the soccer ball, "It's at rest relative to the train and those people sitting within it, but it (like the train itself) is moving 17.8 m/s north relative to the ground outside."

Now imagine that a second train appears in the distance, heading directly toward your train! It's on the same tracks and a collision seems inevitable. The other train is going 20 m/s south. From the point of view of someone in the southbound train, how fast is the soccer ball moving? You can probably work this out. It appears to be moving faster than 17.8 m/s, certainly. A person standing on the tracks in front of the train would measure the ball moving toward them at 17.8 m/s. If they were to run toward the train, the ball would approach them faster. And people in the southbound train would measure the ball approaching them 20 m/s faster than if they were standing on the tracks. They'd measure the ball's velocity as 37.8 m/s northward.

Taking things a step further, imagine that you roll your soccer ball toward the front of the train with a speed of 2 m/s; i.e. 2 meters per second faster than the train itself. That's a pretty natural perspective for you, to measure motion within the train relative to the train itself. It's what you see. Now how fast is the ball moving toward a seated person in the other, southbound train? The ball is approaching them at a rate 2 m/s faster than before, so the answer must be 39.8 m/s. (And if you want its velocity, we'll add a "northward" to our answer.)

People summarize this idea by saying "all motion is relative to some frame of reference", where the word relative means "considered in relation to". That is, whether something is considered to be moving (and if so, how fast it's moving) must be considered in relation to a particular "frame of reference" or "reference frame", or if you prefer, a particular perspective. Is your house moving? It depends on the reference frame. From some vantage points, yes. From others, no.

Why the term "reference frame"? The term suggests an imagined Cartesian coordinate system overlaid on the world, such that the object under investigation can be seen to progress (or not progress) from one set of coordinates (x1,y1,z1) to another (x2, y2, z2) as time passes. Imagine holding a giant pane of glass, with a 2-D coordinate system painted on it, at arm's length. You stand in front of your house and look through the glass at your house, noticing that the doorknob on your front door lines up with the coordinates (8, 12) on your painted coordinate system. You pause a moment and then note, yep, it's still lined up with that same point. From your reference frame, the doorknob (and every other part of the house) is not moving. Then you get into the back seat of a car and the car starts to drive away. Still looking at your house through your reference frame, the house appears to be moving. The doorknob is no longer at (8, 12). Your house has a speed.

Now imagine being able to place your pane of glass anywhere in the world, such that you (or anyone else) can witness any event through it. Whenever you describe motion, you must clearly state where you've parked your pane of glass; i.e. you must clearly state your reference frame. Admittedly, people don't do this in day-to-day living, because it's unnecessary. It's implied (in most cases) that the reference frame is affixed to the Earth. We say, "I was driving 60 mph", and everyone knows what we mean. People may not generally think about it this way, but everyone really knows that your sentence implies you were going 60 mph faster than the Earth (or, better yet, the ground beneath your car).

This idea of relative motion, as it's called, has been around at least since the time of Galileo (1564-1642). He is the first person we know of to speak of it. Yet most people think of Einstein when they hear the word "relative" (in this context). They think of his relativity theories. We'll address Einstein's insights into relative motion soon, but first let's consider what the great Isaac Newton (1643-1727) had to say on the matter.

Newton proposed the following question, which I'll put in my own words: Could you say a rock was moving if it was the only thing in the universe? If you say "yes," then where have you placed your reference frame? If you say the rock is moving at 5 m/s, well ... 5 m/s faster than what? Things can move relative to one another. If you also existed in this universe, and the rock moved toward you, or you moved toward it, then you could state that it was moving from your frame of reference. But if you're not there, and neither is anything else, then the rock can't move toward or away from anything. Newton answered this question with a "yes!" Yes, you can say the rock is moving. He suggested that the rock would be moving relative to absolute space -- space itself. (Not outer space, mind you, but the 3-dimensional space through which we move every day. The space represented with our Cartesian coordinate system.) His notion of absolute space, however, was not well defined. What exactly IS space? Is it a thing? And how do we compare speeds to it?

Newton's contemporary, the great German philosopher Gottfried Wilhelm von Leibniz, opposed Newton on this point. To him, space was not a thing. It was a mathematical construct. Space did not exist independently of the objects in space. Something could not move relative to space itself.

Who was right? Newton or Leibniz?

We now believe that neither man was right. And this modern view is a result of work begun by none other than Albert Einstein (1879-1955).

Einstein's Special Theory of Relativity (1905) and General Theory of Relativity (1915) have much to say about motion, and much of it is far from intuitive. We shall explore the two theories briefly here, but I shall not attempt to explain them fully. (I couldn't do so even if I wanted to.)

In Special Relativity, Einstein made the seemingly outrageous claim that the speed of light will appear the same from ANY reference frame! A woman on Earth would measure the exact same speed for light coming from the Sun as would an astronaut zooming toward the Sun at 20,000 mph. This startling claim furthermore suggests that it is impossible to travel at the speed of light, for then the light from the flashlight in your hand would not appear to be moving away from you at that universal speed we all measure.

What are the repercussions of this crazy claim? And should we believe it?

We'll explore the repercussions by analyzing the follow two scenarios.

In diagram (a), a light source emits a beam of light, which travels to the far mirror and then reflects back to a sensor, placed right next to the source. Perhaps the distance between the two mirrors (L) is one meter. If we choose our reference frame so that the mirrors appear at rest, then we simply see the light travel up one meter and down one meter, for a total distance of 2 meters. In diagram (b), we change one thing, our reference frame. From this new reference frame, the two mirrors are traveling rightward at some constant speed, as is the beam of light (which always stays between the mirrors).  From this new reference frame, the path of the light is now up and to the right, then down and to the right, tracing out two sides of a triangle. Simple geometry proves that this distance is greater than 2L (2 meters); the hypotenuse of a triangle is longer than either side length. So, we've measured a different travel distance.

Now, Einstein says that the light's measured speed will be the same in these two reference frames. Yet speed is defined as distance traveled divided by the time of travel. If the distances differ, yet the speeds are to remain the same, then the times must also differ. And indeed they do. Einstein's conclusion: If light is to have the same speed regardless of reference frame, then both space and time must be relative. That is, the spatial separation or distance between two events depends on the chosen reference frame, as does the time separation between two events. This is BIG, so I want to make sure you understand. To put it loosely, "time flows at different rates in different reference frames". If you and I are moving at vastly different speeds, and our reference frames are fixed to our bodies, then I might measure the time between two flashes of light as 2.7 seconds, while you might measure it as 1.8 seconds. And neither of us is wrong. This is what it means to say "time is relative". We might even say "time is personal". There is no correct elapsed time; there is no universal master clock to which we can point. Furthermore, I might say that the two flashes occurred at the same place, while you might have measured them as occurring 2 meters apart. Again, neither of us is wrong.

Space and time separations are not the same in different reference frames. For light, they generate the same constant velocity when you divide one by the other. But for everything else (with mass), they generate different velocities. Yes, we're already used to observing different velocities in different reference frames (as in the case of the soccer ball), but this theory of special relativity tells us to take a closer look.

Here's the deal. Back to the soccer ball in the train. Let's say the train is traveling north at 17.8 m/s (relative to the ground), and you roll the ball north at 2 m/s (relative to the train). The ball's speed relative to the ground is then 19.8 m/s. We simply add the velocities to obtain this result. But there must be a flaw with this model of the world, because this model could easily lead to a ball exceeding the speed of light, which just can't happen according to our best models of the world. Imagine the train is traveling at 3/4 the speed of light (relative to the ground), and the ball is kicked forward at 3/4 the speed of light (relative to the train). If we were to simply add 3/4 to 3/4, we'd get that the ball's speed, relative to the ground, was 1.5 times the speed of light. Einstein gives us a new model.

u' = the velocity of the object in some reference frame, which we'll call the first reference frame (e.g. ball going 2 m/s relative to train)

v = the velocity of the first reference frame, relative to another reference frame, which we'll call the second reference frame (e.g. train going 17.8 m/s, relative to ground)

u = the velocity of the object as seen in the second reference frame (e.g. the speed of the ball, relative to the ground, which is something like 19.8 m/s)

c = the speed of light in vacuum: 299,792,458 m/s

Notice the fraction in the denominator is tiny, because you're dividing by such a large number (c2). We're adding 2 + 17.8 and then dividing by a number that is practically 1 + 0 or 1, to get the 19.8 m/s we were expecting. But it's not exactly 19.8 m/s. The model actually gives us a more accurate 19.799999999999992 m/s.

But let's see what happens when we model a ball moving at 3/4 light speed (224,844,343.5 m/s), relative to a train, when that train is moving at 3/4 light speed relative to the ground.

That fraction in the denominator is no longer insignificant, and our overall answer comes out to be 287,800,759.7 m/s, which is LESS than the speed of light in vacuum (at 299,792,458 m/s), as it must be.

In general, when is the fraction in the denominator [ u'v / c2 ] going to be significant? Only when one or both speeds is very large, as in a significant percentage of the speed of light in vacuum. When these two speeds are relatively small, we'll simply treat the fraction [ u'v / c2 ] as approximately 0, which means we're adding u' + v and then dividing by about 1.

So here's what we'll do going forward. When we want to work out an object's speed in a different reference frame from the one in which it's stated, we'll see if either the object's speed or the reference frame's speed (relative to another reference frame) is greater than, say, 50,000,000 m/s. If so, we'll use Einstein's model for working out the object's speed in the new reference frame. And if the speeds are smaller than this, then we'll still use his model, but we'll approximate. We'll add the two velocities in the numerator and then divide by 1; i.e. we'll just add the two velocities.

Here's the approximation, spelled out:

Let's practice using this new model

Designed by Freepik

This model of Einstein's assumes that the second reference frame is moving at a constant velocity, relative to the first. It can't be accelerating.

Special Relativity can't handle accelerations. General Relativity can.

Example 1

An airplane is flying eastward at 130 m/s, relative to the ground below. A man walks forward (eastward) within the airplane at 2 m/s, relative to the airplane. What is his velocity, relative to the ground below?

The speeds are small, so we'll simply add u' + v = 2 i + 130 i = 132 i (m/s).

Had we not approximated u'v / c2 as 0, we would have calculated 131.9999999999996 i (m/s).

I'm re-introducing a method for keeping track of direction, covered in detail in a previous section. The i means the motion is eastward or "rightward".

Example 2

An train is heading eastward at 14.0 m/s, relative to the ground below. A man walks toward the back of the train (westward) at 1.5 m/s, relative to the train. What is his velocity, relative to the ground below?

The speeds are small, so we'll simply add u' + v = -1.5 i + 14.0 i = 12.5 i (m/s).

Notice that westward motion is designated by a negative sign: -1.5 i. Stop and imagine the man walking in the train. Picture it in your head. You should be able to justify that his speed must come out to be less than 14.0 m/s, relative to the ground.

Example 3

A spacecraft is headed "forward" (say, toward Jupiter) at 20,000 m/s, relative to the Earth. A missile is shot forward from the spacecraft at 400 m/s, relative to the spacecraft. What is the velocity of the missile, relative to the Earth?

The speeds are small, so we'll simply add u' + v = 20,000 i + 400 i = 20,400 i (m/s).

Example 4

A spacecraft is headed "forward" (say, toward Jupiter) at 150,000,000 m/s, relative to the Earth. (This is not even remotely feasible with current technology). A missile is shot forward from the spacecraft at 5,000 m/s, relative to the spacecraft. What is the velocity of the missile, relative to the Earth?

One of our speeds is above the (somewhat arbitrary) cutoff of 50,000,000 m/s, so we'll use Einstein's complete model:

The equation at right simplifies to 150,003,748. We can say the missile is moving away from the Earth at

150,003,748 m/s.

I left out of the equation an indication of direction, mainly because it's difficult to include it with the equation editor I'm using. We could, if we needed to, designate forward as i, and then attach an i to each number, excluding the "1". The answer would have an i, as well, indicative of forward motion. (This comment applies to Example 5, as well.)

Example 5

A spacecraft is headed "forward" (say, toward Jupiter) at 1,000,000 m/s, relative to the Earth. The spacecraft's headlight is then turned on, and a beam of light races away from the spacecraft at light speed, relative to the spacecraft. What is the velocity of the light beam, relative to the Earth?

You know what the answer is supposed to be. The speed of the light beam should be everywhere and always 299,792,458 m/s (in vacuum). Let's use Einstein's model and see if this is what we get.

Yes! The model tells us that the speed of light is also 299,792,458 m/s, relative to the Earth.

Example 6

Within a particle accelerator (a fancy machine used by physicists) a proton is shot forward at 90% of light speed, relative to the ground. An electron is shot toward the proton at 80% of light speed, relative to the ground. How fast is the proton moving, relative to the electron? (That is, how fast is the proton moving from the point of view of the electron?)

Notice that both speeds are stated relative to the ground. These two speeds are both relative to the same reference frame, but our answer will be in terms of a different reference frame. This is different than previous examples, but we handle this situation using the same model:

90% of c = 269,813,212 m/s

80% of c = 239,833,966 m/s

We use Einstein's model, without approximation.

This simplifies to 296,306,499 m/s.

Should we have used a negative sign in the model, to account for one of the objects moving opposite the direction of the other? No. Intuitively, you should understand that our answer must be larger than 90% of c. This requires a plus sign. The negative sign only pops up when the two original speeds are given in two different reference frames, where one of those frames is moving relative to the other, and where the frames are moving in opposite directions. That's a mouthful. I think the best way to get the model right is to stop and think about what's happening. Try to picture it in your head. You should know in advance whether your answer should be larger or smaller than the speeds you're starting with.

At the risk of doing too much, I'm going to show you a shortcut for solving the problem above.

90% of c = .9c

80% of c = .8c

So our answer is .98837c, or 98.837% of c. And, yes, that comes out to 296,306,499 m/s.

Example 7

A rocket is flying northward at 65% of c (which is not realistic), relative to the ground. A man is riding the rocket (like in that classic Kubrick film Dr. Strangelove or: How I Learned to Stop Worrying and Love the Bomb). He points a gun behind him and fires the gun. The bullet leaves the gun (southbound) at 1,200 m/s, relative to the gun. Find the velocity of the bullet, relative to the ground.

65% of c = 194,865,098 m/s, which is greater than our cutoff of 50,000,000 m/s (which is about 17% of c),

so we won't approximate.

This simplifies to 194,864,405 m/s northward.

Notice the use of the negative sign. Before you even set up this calculation, you should intuitively feel that the speed of the bullet, relative to the ground, should be less than the speed of the rocket, given that the bullet was fired backwards. A negative sign is required to obtain a smaller speed.

Yes, you could have used the shortcut, if you had converted 1,200 m/s to a % of c.

PRACTICE PROBLEMS

So, why are we doing all of this?

The notion that all speeds are relative to some reference frame is a pretty important notion. It leads to the idea that nothing in the universe is at rest, in an absolute sense. It's one of those big ideas that can change how you view the world. When we add Einstein's realization that light is special, in that its speed is the same in all (non-accelerating) reference frames, it can be enough to leave our heads spinning. It also gives us a framework for considering some fascinating questions, like "Is time travel possible?" (Short answer: yes, but only towards the future)

Okay, recall our earlier discussion of Newton and Leibniz. Newton said that a rock, alone in the universe, can be said to be moving or at rest, because we can compare its motion to absolute space itself. Leibniz said "nonsense!" And I said that both men were wrong. How did Einstein disprove both theories?

Einstein did away with the idea of fixed or absolute space. There is no "correct" distance between objects or events. The distance between two objects or events depends on what you're doing when you make the measurement; more specifically, it depends on your speed (really, the speed of your reference frame) relative to the speed of others.

The Earth is a long way from the center of our Milky Way galaxy, some 2.5 x 1017 kilometers. But this distance is from a particular point of view, a particular reference frame attached to the Earth. If you were traveling toward the center of the galaxy at great speed, you would measure a different (smaller) distance to your destination. Furthermore, if you were to actually make the journey, humans on Earth would record one time for you to reach the center of the galaxy, and you would record a different (smaller) time. How long would the trip take according to someone on Earth? It depends on your speed. Assuming you traveled near the speed of light, which is the maximum allowable speed in the universe, it would take you a little longer than 26,000 years, from the point of view of people on Earth. Interestingly, this is NOT long enough to watch all of the video currently on YouTube! Not even close. (source)

Now, while people on Earth would literally die while waiting on you to reach the center of the galaxy, it's possible for you to reach the center in just a few years, from your own reference frame. (The exact travel time depends on your speed.) The passage of time is relative. It's a crazy notion, and we won't really explore it here, but the takeaway is this: If intervals in both space and time are relative, then not only can you not definitively state the speed of something, but you cannot definitively state that it's moving through space.

However, this is not the end of the story. In the word's of Einstein's teacher Hermann Minkowski, "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a union of the two will preserve an independent reality." That union is called spacetime. We live within spacetime. We move through spacetime. Yes, we move through space, and we also move (forward) through time, but competent people can disagree on how much space we move through, or how much time elapses during the moving. But there is no disagreement as to how much spacetime we move through. And how is this measured? Actually, it's rather straightforward. There is a number, labeled the "spacetime interval" that can be calculated, that marks the separation between two events. This spacetime interval can be universally agreed upon. Which makes spacetime an absolute. And so while we can't say that a rock is moving through empty space, we can say that a rock, alone in the universe, is moving through spacetime. Leibniz was off base thinking that space was not real. It IS real, but it is merged with time in a way in which the two cannot be cleanly separated. We live in spacetime. And motion can occur relative to spacetime itself. Wow.

I haven't really spoken of Einstein's General Theory of Relativity. Let's address it briefly. General Relativity (GR) is basically an expanded version of Special Relativity (SR). It's expanded in that the theory models not just motion at constant speed but accelerations, something SR couldn't handle. It also incorporates gravity, and it tells us how to track the motion of objects in a world with gravity. SR was a model that didn't include gravity, which limited its usefulness for describing our world. Interestingly, in GR, Einstein makes the claim that accelerations and gravitational fields are equivalent. We won't discuss this equivalence here (beyond me including Figure 1.3), although it makes for a fascinating discussion.

GR also concerns itself with the structure or shape of

spacetime. In the words of the prominent physicist John

Wheeler (1911-2008), "Spacetime tells matter how to move;

matter tells spacetime how to curve."  That is, the theory

describes how something will move through spacetime,

due to the "shape" of spacetime, while it also describes

how objects within spacetime affect the shape of spacetime.

A popular analogy is to imagine spacetime as a rubber

sheet (depicted at right). Objects with mass, like bowling balls,

cause depressions in the rubber sheet. That is, massive objects

warp the sheet. This warping then affects the motion of other

objects, such as a marble rolling nearby. If the marble

(which makes its own small depression) rolls near the bowling ball, and it's not going fast enough, it will fall into the depression caused by the bowling ball. And it will collide with the bowling ball.

In other words, massive objects cause physical warps in spacetime. Yes, space itself is bent or curved! The Sun, for example, warps spacetime significantly. So does the Earth. If an asteroid (which creates its own little warp) passes too close to the Earth, without going fast enough, it falls into the depression caused by the Earth. It crashes into the Earth.

Thus, what we think of as gravity is actually warps in spacetime.

Okay, so how will we use this newfound information?

Mostly, we won't. I haven't explained either Special Relativity or General Relativity with enough detail that you can actually use these models, aside from calculating relative velocities when dealing with objects traveling near light speed.

The point of all this, again, was to give you a deeper understanding of how our world works. To give you a backdrop against which you'll model motion, typically things like cars and planes and balls on Earth.

"Frame of reference" is an important concept to understand, even if we won't shift between frames much during our modeling. Most of the motion we analyze will be on the Earth, and our reference frame will be affixed to the Earth. We'll talk of a car going 20 m/s, where the frame of reference is implied. It becomes tedious to always state the frame when it's always the same: the Earth, so we typically won't even state it. But realize that you'll use reference frames extensively if you study engineering mechanics in college.

Actually, Newton and Leibniz were interested in whether something could accelerate, not just move, relative to absolute space itself. This was an important distinction for them, but it won't be for us.

Humans can and do agree on distances between places and things all the time (for example, the distance between your house and the nearest gas station) because we are all traveling at roughly the same speed. We're all stuck on the same rocky planet, spinning around and orbiting the Sun. An increase of your speed by 60 mph is insignificant and makes no noticeable difference in any distance you'd measure.

Thus, the relative nature of distances (or space) is rarely seen (except in experiments carried out by scientists).

problems