# Relativity

I first became interested in Albert Einstein and relativity when I was a child. I remember my father telling me about Einstein’s claim that time goes by at different rates and that objects change size for people travelling at different speeds. “That’s ridiculous,” I remember thinking. “It doesn’t make any sense. How could that possibly be true?”

Well, it is true. And one of my pet hobbies throughout my life has been reading popular physics books in an attempt to answer my childhood questions. Coming to an understanding of it – OK, understanding is too grand a word. Let’s say I now accept it. – has not been an easy journey for me. It’s been about 35 years in the making.

Oddly, I’ve always intuitively understood relative speed. I got it from driving in cars. It was always clear to me, sitting in the back of my parents car as we zoomed down Perring Parkway, that our speed, relative to a car driving next to us, was zero; even though our speed relative to someone standing on the side of the road was 65 mph.

The next step in my journey was understanding relative motion. Imagine someone on a long train ride tossing a ball up and down as they cross the country. To the person riding in the train car, the ball behaves perfectly normally; it goes straight up and comes straight down, just as you would guess. However, to someone watching the ball from the ground as the train sped by, the ball would be moving forward as it moved up and down. The ball, therefore, would execute a sine wave motion through space. But no need to stop there. To someone watching the ball from orbit, the ball would not only execute a sine wave but the path of the wave would be bent into an arc, since the surface of the Earth on which the train is riding is curved. Move out another step and watch the ball from outside the Solar System. The motion of the Earth around the Sun comes into play making the ball execute a long, curving series loops (think of a SpiroGraph).

So which of those motions is correct? Einstein’s insight is that they’re all correct, simultaneously. Each is correct for that particular observer’s frame of reference. And no frame may be selected as being ‘more correct’ than any other observer’s frame. As long as we share an observer’s frame, we’ll all see and measure the same things. Therefore, the physics is constant. However, it changes from to frame. It’s unavoidable. All observer’s sharing notes within a frame will report seeing the same thing. Observers sharing notes between different frames will report seeing and measuring different things. Odd but clearly true.

Time and space dilation is more subtle and took longer for me to get a grip on. In honor of this being the 100th anniversary of Einstein’s paper on the Special Theory of Relativity, Astronomy magazine has done a special issue devoted to the subject. An illustration in one of the articles has finally made it clear to me how this works.

Imagine a sailboat passing a dock on which there stands a man. On the 32 foot high mast of the sailboat is another man who drops a rock onto the deck of the boat. From the viewpoint of the man on the mast, the rock drops straight down, taking one second to hit the deck. However, from the viewpoint of the man on the dock, the rock moved at an angle through space onto the deck below. It moved through a longer distance. And this is key, the faster the boat is moving, the longer a distance the man on the dock would see the rock traversing. This is not an illusion. It’s quite real.

And it gets scarier still. Velocity is distance travelled divided by the time it took to cover that distance; i.e., 60 miles per hour. The velocity (speed) of light is constant, 186,200 miles/second. It never changes. So let’s change the falling rock to a light beam from a flashlight. And let’s lengthen our pretend mast to 186,200 miles high and speed up our sailboat so it passes by more quickly. Our friend on the mast turns on his light and it hits the deck straight below him one second later. Our friend on the dock sees the light angle though space and take a much longer path to the deck. He sees the hypotenuse of a right triangle made by the deck, the mast and the front edge of the beam moving down to the deck. Think about this. The speed of light is the same for both observers. The distance the light beam has travelled is longer as measured by dock-man. So if we keep the speed of light constant and increase the distance it travelled how do we make the math work?

This is what the man on the mast measures (C equals the speed of light):
C = 186,200 miles/S = distance / time = 186,200 miles / 1 second

This is what the man on the dock measures:
C = 186,200 miles/S = distance / time = 190,000 miles / 1.02 second

Both time and space are different between observers moving at different speeds with respect to one another. They perceive lengths of distances to be different and time actually passes at different rates for them. Amazing.

Einstein called facing up to the result of that mind experiment “taking the step.” Once he knew the observers would measure different lengths for that light beam he knew that time had to be variable for them. That “step” was the beginning of a revolution in the way the universe is seen to function. Kip Thorne, physicist and author of Black Holes & Time Warps, Einstein’s Outrageous Legacy is still astonished by what he termed Einstein’s “breathtaking arrogance.” His complete confidence that his definition of time and space was correct and everything that had gone before was wrong. Close, yes. But wrong.

Lest you be tempted to think this is a lot of esoteric egghead nonsense that has no bearing on the real world, think again. When the constellation of GPS satellites was designed for US military in the 1980’s, relativity functions were built into the software. The functions were designed so they could be switched off from the ground just in case Einstein was wrong. They’ve never been switched off. Were they to be, users would experience locational errors on the surface of the Earth of about 7 miles. Relativity is completely real.

Congratulations Albert, on the 100th anniversary of your theory. As amazing as it seems, you were right.

## 17 thoughts on “Relativity”

1. The functions were designed so they could be switched off from the ground just in case Einstein was wrong. They’ve never been switched off.

From General relativity in the global positioning system

Neil Ashby

At the time of launch of the first NTS-2 satellite (June 1977), which contained the first Cesium clock to be placed in orbit, there were some who doubted that relativistic effects were real. A frequency synthesizer was built into the satellite clock system so that after launch, if in fact the rate of the clock in its final orbit was that predicted by GR, then the synthesizer could be turned on bringing the clock to the coordinate rate necessary for operation. The atomic clock was first operated for about 20 days to measure its clock rate before turning on the synthesizer. The frequency measured during that interval was parts in faster than clocks on the ground; if left uncorrected this would have resulted in timing errors of about 38,000 nanoseconds per day.
http://www.phys.lsu.edu/mog/mog9/node9.html

2. “…no frame may be selected as being ‘more correct’ than any other observer’s frame.”

Actually, that’s not correct. Right out of Einstein, we find that there are two kinds of frames of reference: “inertial” frames and “accelerating” frames. Special relativity relates to inertial frames, General relativity to accelerating frames.

All inertial frames of reference are valid; none is “better” than any other. (Note that any inertial frame of reference will measure any other inertial frame of reference as moving with a constant velocity.)

All accelerating frames of reference are invalid; none of them are correct. (For instance, C is not constant when measured using accelerating frames of reference. It can be greater or less than the published number, and it can be different in different places.)

Sadly, every frame of reference you cited in your example was an accelerating frame. You said, “So which of those motions is correct? Einstein’s insight is that they’re all correct, simultaneously.” Actually, according to Einstein, all of those particular frames of reference are incorrect. (Alas…)

That insight into the difference between inertial frames and accelerating frames was one of Einstein’s key insights. For instance, it answers the problem of the “twins paradox”. One twin leaves the planet on a high speed round trip tour by space ship; the other stays home. When the traveling twin returns, he will not have aged as much as the one who stayed home.

But why is it not the other way around? During the high-speed portion of the space trip, each twin measured the other as moving at relativistic velocities. If everything is relative, once the traveling twin returns home, shouldn’t he simultaneously be older and younger? No; he’ll be unambiguously younger, and both twins will agree that the traveling twin is younger. Which is to say that both twins will agree that the traveling twin was the one whose time scale was affected by high velocity.

That’s because the twin who travelled spent part of his time accelerating and decelerating. Thus he spent part of his time in an invalid frame of reference. Therefore, his measurements showing the stay-at-home twin apparently moving at relativistic velocities were wrong because they were based on an invalid frame of reference. The stay-at-home twin used an inertial frame and his measurements are correct.

3. A really amazing thing about Einstein’s Special and General Theories of Relativity, and all of the wonderful results that come out of them, is that they both were discovered using a single principle, what Einstein called the Principle of Equivalence.

In the case of Special Relativity, Einstein postulated that all the laws of physics should be the same for two observers in relative motion, as long as they weren’t accelerating with respect to each other. In other words, the two observers are completely equivalent with regard to the laws of physics and, in particular, the laws of electromagnetism, which predict the speed of light to be a certain constant ‘c’. &nbsp Now, once you assume that the two observers will measure the same speed for a beam of light, you have to reconsider how the two observers will measure distance and time. It turns out that, with a little trigonometry, a new set of equations for distance and time can be derived, called the Lorentz transformation equations, and it is this minor trigonometric tweak to the equations for distance and time that cascades through the other laws of physics to produce spectacular results like mass-energy conversion, time dilation, and the cosmic speed limit ‘c’.

In the case of General Relativity, Einstein again started with the Principle of Equivalence. &nbsp He asked, “What if the laws of physics were the same for an observer standing in an elevator in an office building, say the Empire State Building, and an observer standing in an elevator accelerating at 1g through interstellar space?” &nbsp Both observers would obviously experience an acceleration ‘g’ holding them to the floor of the elevator, but Einstein postulated that all the laws of physics would be the same for both observers, ie. that they are equivalent. &nbsp In particular, the laws of electromagnetism again, which, for the observer accelerating through interstellar space, predict that a beam of light travelling from one wall of the elevator to another will appear slightly deflected by the acceleration. If the two observers are equivalent, Einstein then predicted, a beam of light in the other elevator, accelerated only by gravity, should also be deflected. It is this simple thought experiment, and some very complicated mathematics, that leads to the predictions of General Relativity, including gravitational time dilation, gravitational lenses, etc.

I wonder if the Principle of Equivalence will lead to more amazing dicoveries in the future.

4. Michael: “the man on the mast would see the rock dropping straight down”.

No he wouldn’t. He’d see the rock flaring out toward the poop deck.

5. For instance, C is not constant when measured using accelerating frames of reference. It can be greater or less than the published number, and it can be different in different places

Steven, I disagree. You *assumed* an acceleration. I never mentioned any acceleration. The sailboat is moving at a constant speed.

You are correct about accelerating frames. If I were accelerating towards the speed of light, I would measure the speed of a beam of light travelling next to me as being slower and slower.

Also, the speed of light is constant with a medium. If it passes into a different medium, say a denser one, it can slow down.

Michael: “the man on the mast would see the rock dropping straight down”.

No he wouldn’t. He’d see the rock flaring out toward the poop deck.

Tyouth, the only way the rock would flare off in some other direction is if it were acted on by some other force. The rock shares the forward speed of the sailboat and mast-man.

Go read the example of the man on the train again. Within his frame, the train rider experiences the ball going straight up and down, doesn’t he? The observer watching him go by sees the ball moving in a sine wave through space. They measure two distinct motions for the same ball. They’re both correct (within their frames) and neither is ‘more’ correct than the other.

Here’s another example to think about. If you’re riding in a car and toss a piece of paper to the person next to you, does it fly into the back seat when it leaves your hand? No. The paper shares your motion. If you drop it, doesn’t it drop straight down onto the floor? Doesn’t the same thing happen in a plane when your moving at 500 mph? It doesn’t flare off towards the back of the plane, does it?

I guess you could imagine wind forces acting the rock in my sailboat example, so it is admittedly a less than perfect example.

6. Steven, it occurred to me (after further thought) that by saying all my examples took place in accelerating frames you meant they took place on Earth. Gravity is an accelerating force, so technically, you’re correct. However, for purposes of these examples the effect is negligable. I could also point out there’s no place in the real universe where gravity is not felt to some degree.

Einstein liked to imagine his various subjects in sealed boxes (like an elevator car) either floating in space or accelerating in space. But those were imperfect examples as well, since anywhere you go in the universe you are being pulled by gravity, however miniscule the effect. The only thing you can do is imagine away those tiny effects since they’re not having any measureable impact on the basic principles we’re discussing.

7. Michael, actually the biggest source of problems for measurements in accelerating frames of reference is if the frame of reference itself is spinning. That’s the case for someone standing on the surface of the earth, which is why the Coriolis effect was initially so puzzling to classic physicists.

The reason C doesn’t have a constant velocity when measured against an accelerating frame of reference is that the frame of reference induces errors in the measurement. Consider if the frame of reference is rotating at 1 hz. If you tried to measure the speed of a beam of light which was a significant distance away from the origin point, then you could conceivably measure the value as either higher or lower than the published value.

Accelerating frames of reference (including those which are accelerating linearly) produce errors which are both permanent and cumulative to the measurements made by observers in those frames. What’s interesting is that the errors persist even if the acceleration ceases and the frame f reference becomes inertial. High school was a long time ago but we went through this in a special class on Relativity that I took, and it turns out that you can explain exactly things like the twins paradox by calculating the cumulative error caused by acceleration on one or both of the twins (in variations of the thought experiment, such as if they both have spaceships).

8. Michael, yes it seems clear to me you’re right about the velocity of the rock essentially continuing forward with the ship.

I plead early morning thinking and experience (sailing)clouded analysis.

9. Ty, no problem. This stuff is more subtle than it appears at first. For years, I’ve been reading through examples like these and always resisted their implications. I was always saying to myself things like, “Well, it’s just a matter of perspective. That’s not what’s REALLY happening. It’s an illusion.”

We’re not in bad company either. Einstein experienced that denial too. He also thought at first there must be some single point from which you could view an event that gave the ‘true’ description of the event. He decided that the observer who was ‘at rest’ with respect to an event had the proper view and proper description. He looked and looked for that spot but could not find it. That’s because NOTHING in the universe is ‘at rest.’ Everything is in motion. The planets are all spinning on their axes. They’re also continously moving in their orbits around the sun (while spinning). The entire Solar System is in orbit around the center of the galaxy, so it’s flying through space at enormous speed, as a group, held together by the sun, and so on. It’s like being inside a giant clockwork, only the clock is moving too!

So Einstein looked for something he could latch onto. Something he could measure things against. Something that didn’t change. That’s when he realized that ‘thing’ he could measure against was the speed of light. Light has a peculiar property. It refuses to be accelerated. You can put a light source at rest (with respect to you) and measure the speed of light it emits. You can then mount that light on the front of a moving object and measure the speed of light it emits. The speed will be the same. Unusual, eh?

So, using light as his ‘constant’, he did these thought experiments with light, like the sailboat example above. When he realized the two observers would measure different lengths for the light beam, but the speed of light was constant for both observers….well, something had to give. That only leaves time to change (to be variable). That couldn’t be possible, could it? He kept looking for a way around it. There was no way around it. The grade-school level division I did above – simple mathematics – demanded it to change. That was his ‘step’. The one he had such a hard time taking. He said that was the hardest step he took. Accepting that time transpired at different rates in different frames.

What made him truly great, though, was that he didn’t stop there. He knew that step demanded a redefinition of physics. He did that. He worked out an entirely new system. For most things, it gives the same answers as Isaac Newton’s system of physics.

But it also made important new predictions, like E=MC^2. That says, quite simply, that mass and energy are equivalent. They’re the same thing. They’re two different forms of the same ‘stuff’. That you can tranform one into the other. That’s the principle underlying fusion power; transforming tiny amounts of mass into vast quantities of pure energy. It’s the power source of the sun and all the stars.

And it all started with his taking that ‘step’ that no one else could bring themselves to take. And it ended with his working out the vast reams of complex physics and mathematics to create an entirely new system of physics. An astounding achievement for one human being.

10. Question: If this is so, then as an object accelerates time would slow, respectively. At what point would time stop? Is that possible?

11. Michael, Steven, etc.,

Thanks very much for this posting and the comments. I think this is the first time I actually understood (the basics of) the concept of relativity.

12. Lori, not only is it possible, it has happened every time a black hole has formed. (At least from the frame of reference of those outside the hole.) The photons of light which were trapped right at the event horizon of the black hole are forever locked in place. They are always on the verge of escaping the hole’s gravity but can’t quite get out. Think of it as running in place forever. From our frame of reference they are frozen in place at a certain point in time.

From the perspective of the photon though, they’re flying through space like always. They exist right at the edge of the hole’s ‘gravity well’ where space is being stretched (or bent inward) at the same rate the photons are moving (only in the opposite direction). Any photons just below those caught right on the event horizon will experience a stretching of space greater than the photons forward motion, and so will slowly slide down into the singularity at the center of the hole.

In fact, Russian physicist’s term for black holes is ‘frozen stars’, by which they mean that time has frozen on the event horizon of the black hole.

But that’s all from our frame of reference. If you were to travel into a black hole inside a space ship you simply experience flying in. I say ‘experience’ guardedly, since the gravity forces acting on your ship and body would rip you both to shreds. The forces are so intense that even atoms are ripped to pieces. Your mass, however, will be added to the black hole and so increase it’s gravity a tiny bit and it’s event horizon will then grow by that proportional amount.

13. Rahul, welcome to the struggle. It’s not easy for anyone. It seems so damn counterintuitive sometimes. But there you have it; it’s real.

14. i don’t know if that’s really someone named rahul or a joke, but in any case, it’s definitely not the guy that administered the severe beating to you earlier.

15. The IP addresses are on different coasts.

Vile Rahul = NJ
Normal Rahul = CA

16. “Vile Rahul” Vs “Normal Rahul”? Haha…. I’m a regular reader/occasionaly commenter on this site, but haven’t been to NJ in years.

Unfortunately for me, “Rahul” is a very common Indian name.