We remember the guy in the fancy wig³, Sir Isaac Newton. He defined time, velocity, and acceleration as part of his three laws of motion relating to objects experiencing a net force, whether that be from gravity or something else. It turns out that Newton is only part right. Yes, we frowned too when we heard this insanity. How could our measurements of time, velocity, and acceleration change? It depends on the point of view, otherwise known as a frame of reference.

Time Measurements

So, to start off, what is time? No, this isn't a trick question. Whether we picture a digital watch or the grandfather clock in the library, we already have a notion of what time is.

There was a time, after dinosaurs but before the internet, when watches ruled the earth, even though they're a rare breed nowadays.

Let's pretend we have one.

If we look at the imaginary watch now and look at it exactly a minute later, we record the change in time using Δt (delta t). In this case, Δt = 1 minute. All we had to do was subtract the final time from the initial time that we read off the watch. Whether you started at 0 seconds and ended your measurement at 60 seconds, or you started 0:15 sec and ended at 1 min 15 sec, we still end up with a change Δt = 1 minute.

We just used a watch as a clock. The difference between these two is more than philosophical. A watch might not be an extremely accurate clock since it depends on a human being to start and stop the recording of time or note the passage of time with precision. And what if the human was thinking about ice cream? Or that cute human that just walking by? Or both?

It turns out that we may use any kind of clock to measure time. Anything which marks the passage of time in regular intervals will do. Our beating hearts, for example, are ticking clocks, albeit with irregular beats. Vibrations within atoms happen at regular and predictable times⁴, and form atomic clocks. A third example is found in the stars. Very quickly rotating neutron stars called pulsars emit pulses of light at precise intervals.

Frames of References

We defined a frame of reference earlier as a point of view. Let's develop this idea with a thought experiment.

Imagine us running at a rate of 6 mph through a back alley in a neighborhood in the middle of the night. (Exciting, no?) The trashcan of the mean-spirited Gaspard is on the side of the road. According to the trashcan (were it to have eyes) we're running at a velocity of u_x = +6 mph. To ourselves though, by ignoring our own motion, the trashcan whizzes the other way at a velocity of u_x = -6 mph.

A jealous friend named Igor decides to run after us. The reason for his hard feelings is that he can't run as fast as we can, even in our imagination. Poor fool. His velocity is u_x(IGOR)= 4 mph, much less than ours.

He begins to pant and gasp for air behind us as his rate of u_x(IGOR)= 4 mph is 2 mph less than our own. According to Igor, though, our velocity isn't 6 mph since he's also moving. To him, our velocity appears to be u_x = 2 mph until he gives us the chase and stops, at which point he'd measure our speed as 6 mph.

Thus far, we've defined three frames of references: ours, Igor's, and the trashcan's. The "point of view" synonym for a frame of reference isn't too far off.

A frame of reference at rest or moving at constant speed is called an inertial frame of reference. Think back to the law of inertia⁵: an object at rest or moving at constant velocity will remain at rest or moving at constant velocity until it feels some type of force. In the case of an inertial frame of reference, everything within this frame experiences Newtonian physics, or seems to.

Galilean Transformations

We call the transformations that take place between inertial frames of reference Galilean transformations. Yes, after our good old friend Galileo Galilei⁶, an astronomer in the Renaissance who challenged the Catholic Church by saying the Earth was NOT the center of the universe. For this, he was placed under house arrest for the rest of his life. A pretty severe punishment for being right if you ask us.

To properly write out Galilean transformations have a look at the diagram below. Please note that the letter u will be used to designate individual velocities, while v will be kept for the relative velocities between frames of references.



In the above figure, we have a frame of reference S at rest, like the trashcan. Let's pick the other frame S' to be ours as we run with velocity u. In this case, we're making u = v, or our speed is the speed of the frame of reference. After an amount of time t has passed, the trashcan defines our position in terms of (x, y, z), or it would if it was a sentient being.

What about our position according to our coordinate system S'? What we are really asking is how to express (x, y, z) in terns of (x', y', z') and vice versa. We know they're related by a velocity v in the x direction. What's velocity? Displacement divided by time. The distance x is then x = vt.

We're moving in one solitary direction we call x when running in that alley, so our y-and z-coordinates don't change. It's just x we need to work with. Let's hope it cooperates.

In an amount of time t, S' is ahead of S with a distance of vt, so we can write that x = x' + vt. We are now ready to write out our Galilean transformations:

x' = x – vt
y' = y
z' = z
t' = t

Note that the only thing that matters is how fast S and S' are moving relative to each other with velocity v. The frame S could also be moving at a constant speed for all we know, but as long as S' moves at a velocity v faster, then it's all good.

Now we start to understand the term relativity a little better, especially when we look back to our velocities relative to everyone else's when we were racing in the alley.

The first thing we should ask ourselves when we read the term velocity is "relative to what?" Relative to another frame? Relative to another person, or object? Relative to both? Yes, in every day situations, we don't need to specify all this brouhaha. There's always a trashcan—or cop waiting on the highway—measuring how fast we're driving in their frames of reference. Technically speaking, we're at rest in our own car, or going slower relative to the police car chasing us.

Please don't try using that argument to get out of a speeding ticket. It wouldn't go over well.

We can also write down Galilean transformations in terms of velocities u_x and u_x'.

How do we measure velocities for the same person or object in different frames of references?

Let's break it down. An observer in frame of reference S measures the velocity of an object to be u_x. An observer in frame of reference S' measures the velocity of the object to be u_x'. The velocity of frame S' relative to frame S is v.

Now breaking it down in math-speak, velocity equals .We figured that t' = t from our Galilean transformations so Δt' = Δt. But we also figured out that x = x' + vt, so we can write Δx = Δx' + Δ(vt). The relative velocity v doesn't change since we are speaking of inertial frames of references, so put another way, Δx = Δx' + Δ(vt) = Δx' + vΔt.

Going back to , we get . What is ^Δx'⁄_Δt'?

It's none other than displacement over time, or velocity u_x'. We already said that Δt' = Δt, so .Therefore, our new velocity u_x' mercifully simplifies to u_x' = u_x – v.
How do we apply this?

Mean Gaspard is protective of his trash can. Let's say that he gets so worked up that he's running at a velocity of u_x(G)= -10 mph towards us, according to the trash can's frame of reference. We're choosing him to go in the negative x direction, as observed in the upcoming diagram, because he's a negative kind of person. If we stop running, then v = 0, and u_x(G)' = u_x(G). In that case, Gaspard makes his way threateningly towards us at a pace of u_x(G) = -10 mph in our own frame of reference.

However, if we soldier on at a velocity of +6 mph, then Gaspard appears to be running at a velocity of u_x'_(G) = u_x(G)– v = -10 – 6 = -16 mph in our point of view. The relative velocity v in this case is ours since we're measuring Gaspard's velocity according to our frame of reference.



If we did an about face such that v = -6 mph, then Gaspard would run at u_x(G)' = u_x(G)– v = -10 + 6 = -4 mph according to our own frame of reference. He's still gaining on us, in a dark alley, lending an urgency to our understanding of frames of reference.

Common Mistakes

As long as we keep track of which direction the object is moving, just x if it's as we've learned here, we should be in good shape.

Brain Snack

Galileo finally got an apology from the Catholic Church for having him tortured. In the year 2000, that is⁷. Better late than never? Nope, we are still mad about it.