FTL Part 2: The Universal Speed Limit

jonah

It is impossible to travel faster than the speed of light, and certainly not desirable, as one’s hat keeps blowing off.
~Woody Allen

Woody Allen
Woody Allen knew the perils of faster than light travel. (source).

This is Part Two of a multipart series on faster-than-light travel. This time, I’ll describe why it’s difficult to travel faster than lightspeed.

At the end of my last article, I told you that the speed of light is constant, independent of the speed of the observer or the source. If I drive past you at half the speed of light with my headlights blazing, the photons of my headlights will be going past you at 3\times 10^8 meters per second and going away from me at 3\times 10^8 meters per second away from me. This is a really fantastic result! If I, say, threw a baseball at 10 meters per second while running past you at 3 meters per second, you would measure its speed as 13 meters per second. Unlike a baseball, light somehow always goes the speed of light with respect to its observer–no matter how many observers there are, where they are, or how fast they themselves are moving.

If we believe this amazing result (and there are good reasons to), then some rather amazing physics emerge, not least of which is the impossibility of going faster than light.

When he derived special relativity, Albert Einstein thought a lot about trains. Since then, most discussions of this field of physics have used trains for their examples. It’s tradition. This blog will be no exception; without further ado, let’s talk about trains.

Time Dilation And Length Contraction

Imagine that Ada Lovelace is standing by the train tracks when a train passes her. Inside the train, her friend, Charles Babbage is performing an experiment. Babbage uses a flash bulb to produce a flash of light, which he reflects off of a mirror directly above the flash bulb so that it returns and hits a photographic plate next to the flash bulb. Babbage notes the time it takes for the light to travel from the flash bulb to the mirror and finally to the photographic plate. The train car is of height h.

Charles Babbage measures the round trip time of the light
The flash bulb fires. The light travels a distance h to a mirror on the top of the train car, and an additional distance h to the floor of the train car to hit a photographic plate. Charles Babbage measures the round-trip time of the light.

Because the speed of light is constant and Babbage is on the train, the light appears to Babbage to be moving along the tracks with the train. High-school physics tells us that travel time is distance divided by velocity. In physics, we typically denote the speed of light as c and elapsed time as t, so we’ll do that here. The time it takes for light to travel from the floor of the train to the ceiling is

    \[t_{to-ceiling}=\frac{h}{c}.\]

Then the round-trip time is just twice that:

    \[t_{babbage}=\frac{2h}{c}.\]

Now imagine that Lovelace can see inside Babbage’s car and measure the round-trip time herself. Because the speed of light is constant, Lovelace sees the train moving along the tracks at a different speed than the light. The light travels diagonally up and along the tracks to catch up with the train, hits the mirror, then travels diagonally down and along the tracks to again catch up with the train and hit the photographic plate.

 

Lovelace observes Babbage's Experiment
From Lovelace’s perspective, the light must catch up to the train. Therefore, it travels a greater distance to reach the photographic plate. The dotted cars represent the car travelling to the right. (Train engine removed for clarity.)

From Lovelace’s perspective, the light must travel

    \[$t_{lovelace}=\frac{2h}{\sqrt{c^2-v^2}},\]

where v is the speed of the train. Thus,

    \[t_{lovelace}=\frac{t_{babbage}}{\sqrt{1 - \frac{v^2}{c^2}}}.\]

Assuming that the train is not already traveling at superluminal speed, v^2/c^2 < 1, so t_{lovelace} is greater than t_{babbage}. More time passes for Lovelace than it does for Babbage. We call this phenomenon time dilation.

A similar experiment demonstrates that Babbage would measure a longer train than Lovelace would. We call this phenomenon length contraction. 

Velocity Addition

Now that we know how Lovelace and Babbage measure distance and time compared to each other, we can calculate how they would clock the speed of a baseball that Babbage throws in front of the train. They would each measure the time the baseball takes to pass through two points, measure the distance between the two points, and calculate the velocity as

    \[v_{baseball} = \frac{d_{two-points}}{t_{baseball}}.\]

Because time dilates and length contracts for Lovelace, something fantastic happens. No matter how hard Babbage throws the ball, Lovelace will never measure it as travelling faster than the speed of light–so long as Babbage measures it as travelling slower than the speed of light.

This may not seem significant at first, but this is exactly how acceleration works for all bodies. With respect to Lovelace, the baseball starts off at the speed of the train, then is boosted by Babbage’s throw to a new speed.  In other words, it accelerates from the speed of the train to the speed of the train plus the speed of the throw.

A rocket ship works in the same way, except that it uses its rockets to constantly “throw” itself faster and faster. As a rocket ship increases in velocity, a watchtower on Earth will observe diminishing returns. No matter how hard the ship accelerates, if it started at zero velocity, it will never reach the speed of light.

Next time I will discuss how it might (only might!) be possible to get around this barrier and go faster than light.

Omissions of Convenience

I’ve glossed over a big part of special relativity: all of the mathematical formalism. What I presented here was part of Einstein’s treatment of Special Relativity when he discovered it. However, a much more fundamental treatment of special relativity involves what’s called Minkowski space. If we accept that we live in Minkowski space, as opposed to Euclidean space, changes in velocity can be treated as a rotation in the position-time plane. Then time dilation and length contraction fall right out of the equations, appearing as a natural consequence of the rotation. This is a long discussion, but if you all want to hear about it, I’d be happy to write a post in the future.

Further Reading

  • It turns out that Einstein was not the sole inventor of special relativity. A huge number of people contributed (and we give them credit in science, though Einstein was the public figure). Roger posted a timeline of special relativity on his blog.
  • PBS posted an article on special relativity by the Nobel laureate Frank Wilczek. It’s a bit dense.
  • The excellent blog Science In My Fiction has an article on special relativity that describes some phenomena. It does, however, confuse special and general relativity. The discussion of spacetime and mass relates to general relativity. The idea that mass increases with velocity is also outdated. This turns out to be an incomplete description. A more modern description (equivalent to mine) is that your momentum becomes infinitely large as you approach the speed of light, meaning that it will take infinite energy to accelerate you further.
  • This is a description of one of the great experiments confirming special relativity.
  • For the intrepid, I’ll again recommend my introductory textbook.

Contrarian Opinions

There are many online “proofs” refuting special relativity. Most of them are scientifically unsound. In the interest of a good discussion, however, I’m linking to the most fair, unbiased, and compelling discussion arguing against special relativity that I’ve found, on an excellent blog on philosophy.  Ignore the first section, “Faster than Light.” It’s based on an experiment that was later shown to be un-reproducible. (The error was a bad electric connection.) If you all are interested, I will devote a later blog post to a defense of special relativity against this argument.

Questions? Comments? Hatemail?

As always, if you have any questions, comments, corrections, or insults, please feel free to comment or send me an email.

29 thoughts on “FTL Part 2: The Universal Speed Limit”

  1. Pingback: FTL Part 3: General Relativity Lets us Take ShortcutsThe Physics Mill

          1. I do have a question, a bit of a big one though. If a student(such as myself), wanted to go to school for astrophysics, and study topics such as this, would it be better to pursue an education in higher mathematics, before studying more physics?

            My thought process is that since physics is based upon mathematics (Minkowski spacetime, Non-abelian Lie groups, Quat/Octernions, etc…) then the physics would “click” better, and the student would have a better understanding of the underlying framework.

            I’m asking this, because I am considering going back to school for either a double bachelors, or a doctorate in either math or physics; and I am having a personal philosphical “tug-of-war” about which path would be a better starting point.

            I realize this is just a personal blog, but I think you might have some insight on what might be a reasonable way to “get the ball rolling”.

            If it helps anything(personality-wise), I personally enjoy making fratcal art, browsing reddit(/r/science), and work as a data analyst/tester(soo dry and boring).

          2. Well, I can tell you what my experience was. I’m currently getting my Ph.D. in physics. But before that I got a double bachelors in both physics and math. I felt that learning the two at the same time was very helpful. I often found they fed into each other. I can also say that, as a theoretical physicist, my math training has been invaluable. I would not be able to do what I do today without having gotten a degree in math.

            If you’re aiming for physics, I strongly recommend getting a bachelors degree in both fields. At least, that’s what I did. And it served me well.

          3. Yes, that does help! I guess now all I have to do, is get started….oohboy; this will be interesting.

            Did you take both of your bachelors at the same time, or separate?

          5. Sounds like a very beefy course load. Up here in Canada, I’m not sure if double majors are a thing or not.

            I am also considering some training in psychology, along with the math and physics. I think it would be a good blend, especially when you start to consider higher level math, and the amount of philosophy that bleeds into mathematics.

          6. A few courses in philosophy will do you good… especially if you can find one on the philosophy of science. But mathematics will help you understand philosophy. The formal logic used in philosophy is math.

            Another path to take would be single-major in physics, but take as many math classes as you can as electives. The effect would be much the same.

          7. I would try and and take a bunch of elective classes as well, especially over the summer, if possible.

          8. Good. Well, if you go the physics path, take as many math electives as you can. Here are the essential math topics you should know:
            Formal logic (usually taught as part of discrete math)
            Calculus
            Linear Algebra
            Ordinary and Partial Differential Equations
            Group Theory (usually taught as abstract algebra 1)
            Real Analysis (the fundamentals of how calculus works)
            Complex Analysis (how imaginary numbers work)

            If you go into something like particle physics or string theory, you should also try to acquire some differential geometry and topology skills.

            In all of this, it’s also a good idea to do some independent research over the summer… via an honours thesis or an internship in a lab.

            I know it’s intimidating, but it’s definitely possible to cover all of that in an undergraduate program.

          9. I have taken high school math up to calculus, and have done some reading about various calculus modes on my own(ricci, etc..) but I am very rusty on my math, it feels.

            Those all sound like very interesting topics, and I’ll probably look into some differential geometry and topology readings, as I would like to work on string theory, I think it( ST) is fascinating.

            I’m definitely going to look into a tutor of some sort, I know I ‘m going to get swamped on all of this. I want to dive deep into the ocean, but I might need a bit of help learning to swim.

          10. It sounds like you’re well-prepared to start your journey.

            To do differential geometry, you need linear algebra. To do topology, you need real analysis. Differential geometry is based on vector calculus. Topology is the high-dimensional generalization of real analysis.

            If you don’t know any discrete math yet, I’d start there. It’s remarkably easy, mind-blowingly cool, and a good foundation for further mathematical studies. I think the textbook I used was “Mathematics: A Discrete Introduction,” by Scheinerman:
            https://books.google.ca/books/about/Mathematics.html?id=EtcKAAAACAAJ&hl=en

            Oh, and I highly recommend learning some computer programming. I am, of course, biased, because I’m a computational physicist. But some basic programming skills are increasingly becoming important in every aspect of physics… I have friends who do string theory simulations and I’ve done simulations for an alternative to string theory:
            http://www.thephysicsmill.com/2013/10/13/causal-dynamical-triangulations/

            Computer programming also uses and will help you develop a number of important mathematical ideas. AND computer programming is fun! 🙂
            If you’re looking for good mathematical or scientific programming side-projects, Project Euler offers many interesting problems. You can learn a lot from doing it.
            https://projecteuler.net/

          11. Wow, thanks for all the tips Jonah! 🙂

            I will definitely start reading up on discrete mathematics, I can’t say I’ve heard much about it, but it sounds quite interesting. I’ll look around and see if I can find a textbook about the subject, so I can get a bit better feel for the conceptual structure; more so than just reading about it online.

            I’ve bookmarked the project euler site, as well as a learning python book you had mentioned in your opinions about teaching mathematics to younger audiences. Looking forward to playing around with linux, I now have a reason to use my linux distros again, wooo.

            And as you had (wisely) said, I won’t be tackling all of this at once, it will be a slow, uphill climb, but well worth the effort. I am so stoked to be going back to school next year.

          12. You’re most welcome. Keep in mind that all of my advice is based on my own experience. Your results may vary. Good luck! 🙂

          13. I know I’m suggesting a lot of stuff. But you don’t have to tackle it all at once. Indeed, you shouldn’t. It’s just things to try to learn as you work your way through university.

          14. I also have a feeling that you might need to update your blog theme, I think we may have borked the comment section a wee bit.

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