Remember E = mc2? It’s the one equation that you are allowed to include in your popular-physics book (unless you’re George Gamow, who couldn’t be stopped). Mark gave a nice explanation of why it is true some time back, and I babbled about it some time before that. For a famous equation, it tends to be a bit misunderstood. A profitable way to think about it is to divide both sides by the speed of light squared, giving us m = E/c2, and take this as the definition of what we mean by mass. The mass of some object is just the energy it has in its rest frame — according to special relativity, the energy (not the mass!) will be larger if the object is moving with respect to us, so the mass of an object is essentially the energy intrinsic to its state, rather than that imparted by its motion. Energy is the primary concept, and mass is derived from it. Interestingly, the dark energy that makes up 70% of the energy of the universe doesn’t really have “mass” at all, since it’s not made up of objects (such as particles) that can have a rest frame — it’s a smooth field filling space.
All of which is to say that the mainstream media have let us down again. C. Clairborne Ray, writing in the New York Times, attempts to explain whether a spinning gyroscope weighs more than a stationary one, and answers “The weight stays the same; there is no known physical reason for any change.” Actually, there is! The spinning gyroscope has more energy than the non-spinning one. As a test, we can imagine extracting work from the spinning gyroscope — for example, by hooking it up to a generator — in ways that we couldn’t extract work from the stationary gyroscope. And since it has more energy, it has more mass. And the weight is just the acceleration due to gravity times the mass — so, as long as we weigh our spinning and non-spinning gyroscopes in the same gravitational field, the spinning one will indeed weigh more.
Admittedly, it’s a very tiny difference — the energy will increase by an amount proportional to the speed of the spinning gyroscope, divided by the speed of light, that quantity squared, which is really tiny. Nothing you’re going to measure at home. (I’m guessing it’s never even been measured in any laboratory, but I don’t know for sure.) And the article is correct to emphasize that there is no difference in mass that depends on the direction of spin of the gyroscope — that would violate Lorentz invariance, which is something worth looking for in its own right, but would be a Nobel-worthy discovery for anyone who found it.
Given what’s in the body of the response, the intro probably should have stated:
“The measured weight stays the same; there is no demonstrated method of measuring physical reason for any change.”
My html sucks and the part “physical reason for” wasn’t stricken through.
“The measured weight stays the same; there is no demonstrated method of measuring for any change.”
Infuriating!
A technical question: wouldn’t a more sensitive way of doing such an experiment be to do an update of Pound, Rebka, & Snider using right-circularly vs. left-circularly polarized photons?
That would seem to me to allow for more precision than weighing an actual gyroscope, no? Is there any theoretical basis for the results differing between spinning photons and a physical spinning gyroscope?
Any experts around that could answer this?
On the subject of the weight of spinning gyroscopes, it might be worth mentioning the late Prof Eric Laithwaite. An amusing account of his long quest can be found at http://www.combat-diaries.co.uk/diary19/diary19chapter_3.htm
Ellipsis, in general relativity photons of different circular polarizations should propagate along the same paths. The best limit is from cosmology, and the limit on any differential propagation is extremely good.
Thanks Sean.
Your intro, however, leaves out, IMHO, the most interesting part.
What is energy, really?
One can provide two answers.
Answer (1) is that energy doesn’t really exist; what exists is that change in fields with time, and energy is another word for how rapidly a (quasi-stationary) quantum field changes with time — ie E=h_bar omega
Answer (2) is that energy doesn’t really exist; what exists is gravitational curvature, and energy is another word for the 00 term of something (the mass-energy tensor) that ultimately drives spatial curvature.
Putting these two answers side by side reveals some interesting points.
(a) The word energy is ambiguous, in that it’s not clear when it’s discussed whether one is referring to (1) or to (2). This is not a completely trivial distinction. The underlying frequency, in the context of (1) can be either positive or negative (the relevant phase of the complex field can rotate clockwise or anti-clockwise), and this matters because it is this choice of rotational direction that gives us particles vs anti-particles. Taking the absolute value of the frequency as energy is a useful convention that is supported by the details of how each interaction works, but is not fundamental. On the other hand, context (2) insists that we are dealing with something positive definite.
(b) The very fact that this confusion exists is somewhat mysterious and is caused, of course, by the fact that the same engine driving the (very high frequency) temporal modulation of fields is also driving the (low frequency) curvature of space. I’ve never seen a satisfactory argument for why this should be the case, why these things should be linked.
Is the increase in mass gamma for a gyroscope with moment of inertia given by mR^2 where R is the radius of the gyroscope:
gamma = 1/(1- omega/omega max)^2)^1/2
where omega max = c/R
Is the increase in mass gamma for a gyroscope with moment of inertia given by mR^2 where R is the radius of the gyroscope:
gamma = 1/(1- (omega/omega max)^2)^1/2
where omega max = c/R
Regarding: “the energy will increase by an amount proportional to the speed of the spinning gyroscope, times the speed of light, that quantity squared, which is really tiny.”
Don’t you mean: “…speed of the spinning gyroscope, DIVIDED BY the speed of light…”
otherwise the increase in energy would be pretty big, right?
Oops, you’re right; I’ll fix it.
Celestial Mech, for small $latex omega$, the extra mass is something like
$latex frac{Iomega^2}{2c^2}$. Just the rotational energy over $latex c^2$.
I don’t think you can have an $latex omega_{max}$ for rotation like you have a $latex v_{max}$ for linear velocity.
Some guy who posts here sometimes wrote a book that touches on this subject.
I am glad that Sean uses the “old” (relatively old!) definition for mass in relativity! If we define mass according to that “old” STR convention that many of us grew up on [ m = (m_0)(1 – v^2/c^2)^(-1/2) ] then the classic E = mc^2 becomes an accurate formula for the total energy we can derive from a piece of matter, whether at rest relative to us or in motion. The classic formula also tells us the effective inertia of the mass-energy concentration, as noted here. Hence, that convention is better despite the tendency now to refer to “mass” as an invariant (i.e., many now say “mass” for the quantity formerly referred to as “relativistic mass.”) The new definition has the usual snooty appeal to certain kinds of idealism, but: it is hard to keep that definitions straight because of ambiguities in how to define “velocity of the mass” in question, even with relative standards.
Regarding the example of the gyroscope, that could be in a more extreme case a flywheel inside a container. The rim (with most of the mass) of the flywheel rotates (idealization) at say r*omega = 0.6c. So, its effective mass (for defining either energy, or the weight the whole thing has on the ground, etc.) is increased to the relativistic level of 1.25. But the flywheel is inside a housing and so I don’t see that, so I am tempted to say that the velocity of the “object” is “zero.” That’s one of the problems with the new invariant definition, it is misleading and ambiguous when different parts of “a thing” move at different speeds. With the old gamma factor, you just add up all the relativistic mass. Then the total is consistent, and a proper measure of total energy and total inertia (effective weight for “real gravity” or acceleration by applied force.)
The inertia of energy, has consequences for the credibility of a universe which has two dimensions of space. The potential for moving say charges together from infinity in such a space is infinite,, so the inertia of their potential energy should be infinite at any separation. This makes physics in 2-d spaces absurd, yet it neither stopped A. K. Dewdney, the author of “The Planiverse, from considering that a plausible physics, nor others writing about things like black holes in 2-space etc. I wonder why?
Andy S., pls. tell me how to imbed symbols/equations in my comments?
tx
PS: Interesting to note, as far as “measured” goes, that no one has yet directly measured Lorentz contraction, true? Isn’t that ironic, with all the complaints about the failures of pseudoscience and etc? Sure, LC makes logical sense (just consider the need to be consistent with time dilation when one body passes another – the alternative is unequal approach velocities!), but to never have been actually measured … If this was something really controversial, we’d hear flack about it!
I much prefer the modern convention in which mass is an invariant. So I would say E = gamma mc^2. In this convention the mass of the gyroscope would stay the same, but its energy would increase due to the gamma factor.
Don’t you have to make a relativistic correction to various thermodynamic equations to take into account the greater mass imparted to a (very) hot gas by the motion of its component molecules?
This reminds me of a story I heard of someone who apparently asked Apple customer service if his iPod would get heavier as he put more songs into it. The question is laughable at first, because songs are just information — surely information doesn’t weigh anything!
But that got me wondering: would the iPod, in fact, get heavier? It seems conceivable to me that its weight might change, since (according to my extremely skimpy understanding of hardware) the information in the iPod is stored in the form of spin states, and these spins might get flipped as a result of adding songs, possibly changing the net energy of the iPod (e.g., from classical electromagnetism the energy of a magnetic dipole is lower if the dipole is parallel to the magnetic field than if it’s anti-parallel.) But I don’t really know in detail how iPods work. Could someone illuminate this to me?
It would be funny if iPods do actually get heavier when you add songs to them. It’d be even funnier if whether or not the weight changes depends on the kinds of songs you add, so for example, Beethoven might cause your iPod to get heavier whereas the Beatles might cause it to get lighter. The Beatles would then have a negative “iPod mass”! Perhaps the possible negativity of the iPod mass could shed light on the nature of inflation and similar problems in physics… in fact, I’m sure Apple is already working on such a product: the i-Cosmology.
What kind of warnings must be given to the users of these iPods?
Nicholas — my guess: maybe. If you think of the hard drive as a bunch of little bar magnets. As long as they cooled the material it was made of slowly enough, it will be in a rather low energy state — the bar magnets will all be anti-aligned.
As you add songs, of course, you start flipping that bar magnets around — most likely out of their minimum energy state. So the energy of the system goes up, and when you divide your iPod by c^2, yes, it is more massive. My guess is that music has a rather high — near maximal — entropy, especially after being compressed as an mp3.
This doesn’t hold if the hard drive is “zeroed out”, or if the drive material was cooled too fast (not properly annealed.) In the latter case, the bar magnets had very little correlation to begin with (i.e., high entropy) because they were frozen in at a very high temperature.
Correction – I should have said
” … (i.e., many now say “mass” for the quantity formerly referred to as “proper mass.”) ”
I think the iPod thing about increased mass is cute. How about human brains?
Also: gyroscopes spinning and having weight. What’s the amplitude of the general relativistic spin-spin coupling of the gyro-Earth system? It should also come in at order c^2, but instead of two factors of the gyro velocity, it’s one gyro, one Earth.
Iblis, I clicked on a real physics link (http://www.claymath.org/millennium/Yang-Mills_Theory/ ) from that joke computer warnings page, and got this interesting scoop that I didn’t know about:
The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the “mass gap:” the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap and will require the introduction of fundamental new ideas both in physics and in mathematics.
What’s up with all that, and why don’t we hear more about it? tx
http://www.springerlink.com/content/q28tw1m14643737v/fulltext.pdf
Hmm. I mean, totally different set-up, and totally different domain of validity, but it does indeed go as
GMM’ww’/c^2
or
GMM’v_E v_G / r_E r_G c^2
which is, on the face of it, quite a bit larger than
GMM’ v_G^2 / r_E^2 c^2
given the smallness of r_G compared to r_E, and v_E is about 100 m/s (a possible gyro speed.)
Sean, you wrote the book on this. Is it possible that the relativistic spin-spin coupling could dominate over the gravitation of the kinetic energy?
bigvlad, I was wondering the same thing. But even in the modern convention, where mass is invariant so we have E = gamma m c^2, doesn’t gamma depend only on the translational velocity of the centre of mass? In the case of the gyroscope if you are in it’s rest frame (i.e. where it’s spinning but not translating), so gamma is 1 but as Andy S. pointed out, the $latex frac{Iomega^2}{2c^2}$ term should still contribute to the mass.
I’m not sure I understand this, though. Does it work like this: gamma m c^2 includes the translational kinetic energy, so the E must include all other forms of energy (potential and rotational kinetic energy)?