Credit: Herb Thornby

As you may know, I’m teaching a General Relativity course this term, and so I have spacetime on the brain. So it’s particularly fun when I get relativity-oriented emails. This morning, for instance, I found a question from a reader named Dawn who asks:

Okay, the info out there seems to be all about the effects of speed and mass on time e.g. The Twin Paradox, but I just can’t see the why.

Take this rather interesting question I came across… ‘If you could spin a carousel fast enough to get its rim moving at nearly the speed of light, would time stand still for people on the carousel?’. So, here, in theory, time would go more slowly for the people on the carousel. Fine. Buy why? And I don’t mean mathematically why, I mean physically why.

For time to be affected by speed and mass it must be a ‘thing’ (even if ultimately time may not exist). I have yet to see an animation, model or drawing that shows WHAT is PHYSICALLY happening to this thing called ‘time’.

In the same way that we see how atoms and molecules are affected by heat and then understand why things get hot or cold. What is physically happening to the ‘atoms’ of time when they are being subjected to speed or mass? I would be particularly interested to see this in the spinning carousel example. Being able to affect time without traveling.

I linked the answer to the original question from the HowStuffWorks website for the rest of your edification, but I’m not surprised that Dawn came away from it with more questions than answers.

For those of you who aren’t familiar, let me give you two basic results from relativity:

1. Moving observers apparently have slow running clocks. The closer you move to the speed of light, the greater the effect. This shows up, notably, in the twin paradox that Dawn mentions in her question.
2. Likewise, clocks run slow in strong gravitational fields. Clocks on earth, for instance, run slower than clocks in deep space by about 1 part in a billion.

Part of Einstein’s genius was that he connected these two effects, and noted that functionally, there’s little difference between an observer who’s moving quickly (at the edge of a Carousel, for instance), and one who’s on the side of a hill. In both cases, you’d be pushed outwards (downwards). I even did a technical blog post wherein I worked out the details for you. Up top of the page, I’ve included a much nicer illustration from my upcoming book to illustrate the basic idea.

But all of that is the “what?” and Dawn, at least, came in already knowing that. Her question more concerns the “why?”.

I’m sorry, Dawn, but I think you’re going to find my answer unsatisfying. The reason is that time (unlike an atom) isn’t a thing. To be blunt, an atom isn’t really a fixed thing either, but I think we should probably leave that level of abstraction for another day.

Let me give you a quick exercise to put your mind in the ride state for what comes next.

I stand up and face North, and having done so, I can now describe the objects around me in various ways. There is a chair about 3 feet to my right, a computer monitor 1 foot in front of me, and so on. These coordinates seem like a perfectly valid way of describing things until I decide to pivot.

Suppose I turn 90 degrees to my right. “Forward”(North) becomes “Left.” “Right” (East) becomes “Forward.” and so on. It would have been ludicrous for me to think of the forward direction as something concrete because I can easily turn and make it some other direction.

Put another way, you’re asking the wrong question.

In a previous post, I tried to tackle this question in a geometric way. The idea is that simply by turning, the coordinates that we might label as “x”,”y”, or “z” get switched around but something — in this case, the distance between any two objects — remains the same.

Time is more complicated because we feel intuitively (and wrongly) that it is somehow wholly different from the 3 coordinates of space. While it has a slightly different behavior, the reality is that space-time is really the thing that is unchanged as we turn around or fly through it at high speeds.

In other words, nothing is happening to time as you travel close to the speed of light. It was, and remains, inextricably combined with space, and you should simply think of the whole operation as looking at spacetime at another angle.

-Dave

My latest column is up, and it’s a fun one. Does antimatter have antigravity? Only one way to find out for sure!

In other news, I’ll be on a panel at the Library Journal Day of Dialog May 29 (NYC). Richard Dawkins, Simon Winchester, and I will be talking about “The Art of Science Books.” It should be pretty awesome.

Lots of excitement in the last couple of months before the new book comes out!

-Dave

Our first review is out, and it’s very good! A few choice quotes from our Publisher’s Weekly review:

…An informative, math-free, and completely entertaining look at the concept of symmetry in physics… Throughout his fascinating discussion, Goldberg’s writing remains accessible and full of humor…Seasoning his expose’ with pop culture references that range from Doctor Who to Lewis Carroll to Angry Birds, Goldberg succeeds in making complex topics clear with a winning style.

-Dave

Credit: New Scientist

Enough with the random announcements! It’s time for some science! I have a new column up on io9: Will the Universe End in a Big Rip? For those too impatient to read through, the answer is “maybe,” with a side of “probably not.” But read it anyway. There’s some good cosmology in there.

-Dave

Okay, just one more, and only because it’s pretty insane. Danica McKellar (who you may also remember as Winnie Cooper from the Wonder Years) has written a really nice blurb for the Universe in the Rearview Mirror:

This is a fun and fascinating examination of core physics concepts, explained with humor and levity – and which even includes a look at one of physics’ unsung heroines, a giant upon whose shoulders many physicists have stood: Emmy Noether!

—Danica McKellar, actress and New York Times bestselling author of Math Doesn’t Suck

In other news:

• I’ll be doing a reading and book signing on release day (July 11) at the Rittenhouse Barnes and Noble in Philadelphia at 7:00pm. Come by and say hello.
• Discover will be reviewing my book in their July/August issue (out June 11). Check it out!
• I should have a new “Ask a Physicist” column out later today at io9. It’s all about the Big Rip!

Exciting stuff!

-Dave

Okay, I promise not to overdo it, but in the last few days, some really awesome people have been saying some really awesome things about my upcoming book, The Universe in the Rearview Mirror:

Most physics books can’t really be described as `rollicking,’ but most physics books aren’t written by Dave Goldberg. This book is fun, irreverent, and enjoyable, but also very truthful and illuminating. Buy it for your friend who was always scared of physics, especially if that friend is yourself.

- Sean Carroll, theoretical physicist at Caltech, author of The Particle at the End of the Universe

Reading this book is like taking a class with the most awesome science professor ever. Goldberg answers the physics questions you secretly want to ask, like whether you’ll ever have a TARDIS and what would happen if Earth were sucked into a black hole. You’ll have so much fun finding out that you won’t realize that you’ve just learned how space and time work at a fundamental level. A must read for anybody who wants to understand the nature of the universe — with jokes.

- Annalee Newitz, editor and time distortion field operator for io9.com, as well as the author of the upcoming Scatter, Adapt, Remember: How Humans will Survive a Mass Extinction

Whether unveiling the mysteries of the Higgs boson, visiting Antworld, or cracking the kaon koan, Dave Goldberg’s masterful explanations of how symmetry shapes the universe will enthrall and enlighten.

~J. Richard Gott, Professor of Astrophysics, Princeton University, and author of Sizing Up the Universe: The Cosmos in Perspective

I’m extremely flattered and touched by their kind words. Please do them a solid by showing their books some love.

-Dave

So I found this in my inbox:

Unputdownable! This book is tremendous fun for any reader curious about our bizarre and beautiful universe. If only the profound concepts and laws of physics were presented in schools in the clear and fun way Dave Goldberg has in this book, we would attract many more people to science early.

Departments of Astronomy & Physics
Chair, Womens Faculty Forum, Yale University

Wow! I mean, right?

Full disclosure, Priya is a good friend and mentor, but she also doesn’t pull her punches. She even makes me want to read the book again!

-Dave

P.S. If you haven’t already done so, be sure to become a fan on facebook. There’s a lot of good stuff there, including talk announcements, links to articles, discussion of ongoing science and more!

Like many of you, one of my earliest and best exposures to popular mathematics writing was John Allen Paulos’s excellent Innumeracy and Beyond Numeracy. He and I have become twitter buddies, and he graciously agreed to blurb my upcoming book:

The scope of Dave Goldberg’s The Universe in the Rearview Mirror is almost as vast as the physical universe it does a most impressive job of describing. Employing an engagingly informal and often humorous voice, he explains some very profound physical ideas, ranging from the the Second Law of Thermodynamics and Maxwell’s demon to Olbers’s paradox of the dark night sky and the mysteries of quantum entaglement. Perhaps most importantly he limns the under-appreciated work of Emmy Noether whom Einstein described as “the most significant creative mathematical genius thus far produced since the higher education of women began.” Her principle that every symmetry gives rise to a conserved quantity unifies much of physics and Goldberg makes clear why and how.

It’s a pretty big honor. If you’re not clear on why, go back and read John’s books, and give yourself a treat.

-Dave

Look what just came out in the cradle of democracy! If you’re an actual Greek person with an actual Greek version of the “User’s Guide,” I would be much obliged to see a genuine photograph.

Sincerely yours,

-Dave

Lately, my my inbox hath overflowed. Yesterday, I got a thought-provoking email from a fellow physics instructor. It’s a good, non-intuitive question about special relativitiy. It’s also got the “technical” tag, so if you’re afraid of a few equations and some truly terrible MS Paint figures, this may not be the right blog for you.

Since you might not make it to the end, let me remind you now to like my new facebook page!. Our correspondent asks:

A box has a mass m. Push on the Box and it has an inertia proportionate to m. If by various processes, some of the matter is converted to energy .. Say by burning fuel or mechanical to electromagnetic or radioactive decay… But the energy is still contained in the box. Does the box have the Same inertia? If answer is simple yes by $E=mc^2$, how does Energy have inertia?

I’m going to rephrase this a little bit for concreteness. Suppose you had an essentially massless box, and inside there was 5 kg of matter and 5 kg of antimatter, separated by a magnetic field or some other such connivance.

The total device, of course, would have a mass of 10 kg by any measure you wanted to consider. Pushing on it with a force of 10 N, for instance, would cause it to accelerate at:

$a=\frac{F}{m}=\frac{10N}{10kg}=1 m/s^2$

Likewise, were you to measure the gravitational pull of the box (which would be tough but, in principle, doable), you’d find it has a gravitational mass of 10 kg.

No problem so far, but what happens when you remove the membrane, and that 10kg of mass turns into:

$E=mc^2=(10kg)(3\times 10^8m/s)^2=9\times 10^{17}J$

worth of photons. Photons are individually massless particles, so the question is, does your box still have inertial mass?

Yes. And it has gravitational mass, too.

To understand why we need to delve a little into special relativity, and in particular, into the postulates of special relativity:

1. The laws of physics are the same in all inertial frames of reference.
2. The speed of light in free space has the same value c in all inertial frames of reference.

This setup is not that dissimilar to how Einstein derived $E=mc^2$ in the first place. So imagine (for simple mathematical convenience) that the light in your box were monochromatic, and the box is stationary, with half of the light traveling to the right, and half to the left.

Light does carry momentum, as we known since Maxwell, and can easily be seen in a radiometer:

$p=\frac{h \nu}{c}$

where $h$ is the Planck constant, and $\nu$ is the frequency of an individual photon. In this case, the momentum cancels.

But now look at the box from a different inertial perspective, one where the box is moving to the right at v. This speed can be much less than the speed of light, and will still produce an interesting answer.

The 2nd postulate of special relativity tells us that all photons travel at the same speed. The only thing that changes if you look at them in a moving frame is their frequency (or equivalently, their wavelength). The frequency of the photons in the forward-going direction are higher than they would be if the box were at rest (blueshifted), and the backward-going direction are _lower_ than they would be if the box were at rest (redshifted). The relation is:

$\frac{\nu_f}{\nu_0}=\sqrt{\frac{1+v}{1-v}}\simeq 1+v/c$

and similarly

$\frac{\nu_b}{\nu_0}\simeq 1-v/c$

So the total momentum of the forward going photons are:

$P_f=\frac{N}{2}\frac{h\nu_f}{c}=\frac{N}{2}\frac{h\nu_0}{c}(1+v/c)$

$P_b=-\frac{N}{2}\frac{h\nu_b}{c}=-\frac{N}{2}\frac{h\nu_0}{c}(1-v/c)$

$P_{box}=\left(\frac{Nh\nu_0}{c^2}\right)v$

Feel free to check my algebra, but the upshot is that there are two terms each in the forward-going and backward-going momenta, and one of them cancels, and one of them adds.

This means that the impulse required to push the box is the value above, and since the box is moving at non-relativistic speeds, we can re-write this as:

$P_{box}=Mv=\left(\frac{Nh\nu_0}{c^2}\right)v$

The bit in the parentheses is the mass. It’s also worth noting that:

$E_{tot}=Nh\nu_0=E$

so

$P_{box}=\left(\frac{E}{c^2}\right)v$

So yes, a collection of photons has inertial mass because it requires an impulse to increase their net momentum.

As a final bonus: does a box of photons have gravitational energy? Absolutely yes! I’m not going to prove this in detail, but I’ll simply give you a flavor for why.

1. The equivalence principle of general relativity says that there is no distinguishing between being in free-fall and a true gravitational field. As a result, all massive bodies fall with the same acceleration in the same field. After all, the curvature describes the acceleration, not some inverse square law.
2. But Newton’s 3rd law really does hold. It gives rise to conservation of momentum, which means that if my box of photons is accelerated toward the earth, Newton #3 says that the earth must be accelerated toward the box with the same force.

Tada! A box of photons has mass even though each individual photon is massless!

Of course, this shouldn’t be such a big surprise. After all, what is the Higgs but a way of turning interaction energy into mass? For that matter, would it surprise you to learn that protons are about 50 times more massive than the quarks that make them? The rest is all interaction energy.

-Dave