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.
ack — clearly the spin-spin coupling can’t be independent of distance between the gyros. Looks like there has to be a dimensionless factor of something like (r_G/r_E) (comparing with the dimensionless factor in the paper linked above — spin-spin force declines as 1/r).
But that would still leave the spin-spin coupling relevant at the level of the gravity of the kinetic energy.
I’m certainly using the “modern” terminology, in which mass is invariant. (By which we mean “invariant under Lorentz transformations.”) That’s why I wrote “the energy (not the mass!) will be larger if the object is moving” above.
Simon, I don’t know about the spin-spin coupling. GR and SR effects are numerically comparable for GPS satellites, so maybe.
But Sean, if we use the “old” usage of relativistic mass to refer to plain “m” when not zero-subscripted (and I do follow the recent history of science terminology to some extent, that is what most writers used to say) then we can use “mass” in a way that actually makes this sentence of yours consistent:
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.
That doesn’t really square, pardon the pun, with what you said after that. The relation m = E/c2 is only true in general if we *do* include the energy of motion etc. That is the whole point of my own illustration about the extra “mass” in the gyroscope, the “rest frame” of the gyroscope as a whole does not follow the motion of the rotor – hence, the “mass” of the gyroscope as a whole is higher despite the *sums* of the “rest masses” being the same. That is just confusing, but summing “relativistic masses” always gives the same number in a given reference frame.
But OK, for convention we can just call the augmented quantity “relativistic mass,” and you can reserve “mass” for the “rest mass” if you want. However, considering the internal motions and mass deficits etc. involved in talking about matter, I consider the latter concept to be awkward. Also, we might as well keep the zero subscript just to be sure what is being referred to.
Simon, it’s pretty easy to show that the spin-spin energy will be quite small compared to the kinetic energy associated with the rotation. Forgive the lack of latex in this little comment, this may end up looking ugly.
Working in units where G = c = 1, the spin-spin energy is (up to factors of order unit, and neglecting angular factors)
E_ss = (Searth)(Sgyro)/r^3
where Searth and Sgyro are the spin angular momenta associated with the earth and with the gyro. The correct value of r to use here is the radius of the earth; this formula comes from considering the torque that acts on a gyroscope due to the “gravitomagnetic” frame dragging field of a spinning body. Note the resemblance to a magnetic dipole sitting in a magnetic field; note also that this interaction energy falls off quite rapidly with distance. (Since in G = c = 1 units angular momentum is a length squared, this formula is dimensionally consistent.) It’s this one over r cubed that is going to kill this. The kinetic energy associated with the gyro can be written
E_gyro = (Sgyro)^2/(2 Igyro)
where Igyro is the moment of inertia of the gyroscope. The ratio of these two formulas is
(E_gyro)/(E_ss) = (Sgyro) (Rearth)^3/(Igyro Searth)
(dropping the annoying factor of two at the level of precision with which we’re making this estimate). Searth = Iearth Omegaearth, and the moment of inertia of the earth is Iearth = (factor that depends on the concentration of the earth’s mass ~ 0.3 or so) times Mearth Rearth^2. Also, Sgyro/Igyro = Omegagyro.
Putting all this together, we have
(E_gyro)/(E_ss) = (Omegagyro Rearth)/(Omegaearth Mearth).
Since we can easily make the gyro spin faster than once per day, and since Rearth/Mearth ~ 10^9 (in G = c = 1 units), the spin-spin interaction is pretty small compared to the kinetic contribution.
Hey Scott — something odd here for me in your derivation? It’s not the energy associated with the spin-spin coupling, but the force itself. i.e., your calculation seems to assume that the spin-spin force is caused by the gravitation of the energy associated with it. Which seems strange? (For example, the potential energy of the gyroscope does not lead to much of an additional gravitational force — it’s the gradient of the potential field itself that dominates!)
Here’s what I would say (I agree with you on the magnetism analogy, BTW — I am not sure why the paper I linked found a 1/r dependence for the force, I agree it should be 1/r^4.)
F_ss = (Searth)(Sgyro)/r^4
F_gyro = Mearth(Sgyro)^2/(2 Igyro)/r^2
So
F_ss / F_gyro = omegaearth / omegagyro
OK, so I agree — highly subdominant. Also, it is very stressful to do simple algebra in front of millions of people.
Doesn’t that depend what you are prepared to accept as a measurement? For example, interferometric techniques provide the most accurate length determinations there are, so it might reasonably be argued that the Michelson-Morley experiment was a very direct measurement of Lorentz contraction.
Personally, I still can’t get over the notion that if you measure a magnetic field due to current in a wire, you are actually measuring the electric field arising from the differential in Lorentz contraction between the positive and negative charges in the wire, due to the mm/s drift velocity of charge. So, anyone who has ever done high school physics has measured Lorentz contraction arising from snail’s pace velocities.
Scott (#29) and Simon (#30), or anyone come to that, in the context of GR if two rotating rigid bodies are in mutual orbit in space, unaffected by other masses, and their rotation axes ae skewed, will GR effects tend to evolve their rotation axes over time to be anti-parallel?
I realise this would be a miniscule effect, if it existed at all, and probably dwarfed even by the small tendency to spiral towards each other as energy dissipated from the system by gravity waves. But I’d be very interested in a sketch of the kind of calculations that might be involved or a reference.
(Also, let’s assume the bodies’ rotation rates are each synchronized with their orbital speeds, or that they are black holes say, so that energy is not lost from the system by thermal radiation caused by distortions changing the bodies’ shapes as they rotate.)
Cheers
P.S. On the off-chance you might have a comment, but prefer not to wrestle
with TeX notation here or risk remarks open to challenge, my email is jhnrmsdn at yahoo dot co dot uk, although obviously a reply would ideally be better here for everyone’s benefit.
Also, if I receive any relevant emails on this (small chance, but one lives in hope!) I’ll summarize them anonymously here, unless the sender explicitly consents to their name being cited.
hi nicholas (#18),
i actually would expect the ipod to get lighter when you store songs. since the process of storing songs involves energy, a fraction of which will be dissipated and radiated off, the net mass will be lower afterwards.
that is if you ignore such huge effects like the fingerprints on the button when you initiate the recording or the associated material loss (paint e.g.), that i would estimate to be at least 10 orders of magnitude more important.
niel B (#23),
that is one of the millenium challenges. i don’t know in what respect you can expect to hear more about it, since it is a purely formal mathematical problem.
Hi Simon —
I think we were just calculating slightly different things. I wasn’t thinking forces, just energies. For the weight of a gyroscope, I agree it’s the force you want, and is certainly the relevant quantity. Sorry to get off topic … chalk it up to blog scanning just before going to bed.
Note that the spin-spin energy I was looking at does indeed lead to the 1/r^4 force you calculated; however it also causes a 1/r^3 torque which causes gyroscopes to precess (torque ~ dE_ss/dAngle). One can interpret the GP-B frame dragging precession as due to this torque — the gyro tries to realign to minimize the interaction energy.
John Ramsden, these couplings lead to precession, but do so in such a way that the *total* angular momentum remains conserved. In other words, you can define J_total = L_orbit + S_1 + S_2, and you find that spin-spin (and spin-orbit) tend to change the direction, but not the magnitude, of S_1 and S_2. In order that J_total be constant, L_orbit “recoils” to compensate. One never finds that the orbits become antiparallel; the components of S_i parallel to J_total remain fixed, while the components perpendicular to J_total precess around. Note that some evidence for these precessions has been seen in binary pulsar systems.
Whoever said Lorentz contraction hasn’t been observed … well, time dilation is observed all the time. Certain cosmic rays (e.g., muons) that are produced in the upper atmosphere should decay before reaching the ground. But, they easily make it thanks to their dilated lifetime. Even more extreme dilation is observed in particle accelerators every day — fast moving particles cover a distance that is set by their dilated lifetime. In the frame of the particle, the lifetime is just the rest lifetime, but the distance is length contracted, so everything is consistent. (Even more prosaically, the existence of magnetic forces from electric charges is a simple consequence of length contraction — see chapter 5 of the E&M textbook by Purcell.)
In response to Mike M — is it not correct that Lorentz suggested the contraction to EXPLAIN the result in the Michelson-Morley experiment? Thus by definition the Lorentz contraction was demonstrated by that experiment.
would a rotating galaxy’s increase in mass affect
dark matter calacultations?
would a rotating galaxy’s increase in mass affect dark matter calacultations?
No … the rotation affects things by a fraction (vrot/c)^2, which is like a part per million or so. To explain galaxy rotation curves you need a factor of several change in the mass estimate. (It’s also worth noting that rotation curves are just one of many lines of evidence suggesting the need for dark matter. Even if one could cook up a model that explained galaxy rotation, you’d still need to explain galaxy cluster dynamics, x-ray temperatures of bound gas in clusters, gravitational lensing, and the amplitude of acoustic peaks in the cosmic microwave background … and probably a few other things that I’m forgetting.)
Whoever said Lorentz contraction hasn’t been observed … well, time dilation is observed all the time.
For a real-life example of Lorentz spatial contraction rather than time dilation, consider the RHIC experiment. Their models depend on the shape of the interaction region where two nuclei collide, so it’s important to remember that the normally spherical nuclei will be flattened into pancake shapes at relativistic speeds.
Spinning Earth, tidally-locked moon, both in free fall around the sun. No spin Nordtvedt effect. No free fall spin effects in Gravity Probe B’s two sets of opposite spin-paired gyroballs. No orbital weirdness for mismatched binary pulsar PSR J0737-3039A/B (arxiv.org/abs/astro-ph/0609417). No vacuum dispersion, dichroism, or gyrotropy for linearlyly polarized light (50:50 mix of left and right circular polarizations) over millions of lightyears path through vacuum.
Is the vacuum isotropic? To EM, yes! (arXiv.org/abs/0706.2031). To physical spin, gravitational binding energy, composition, quantum particle and orbital angular momenta (http://www.npl.washington.edu/eotwash/publications/pdf/prl97-021603.pdf), magnetism, superconductivity… yes! To better than parts-per-trillion relative.
However… given chemically identical opposite parity mass distributions a first order Equivalence Principle violation arises from their diastereotopic interaction with a massed sector chiral vacuum background (opposite shoes fitted onto a left foot) in teleparallel gravitation with Weitzenböck spacetime torsion (transforms like Lorentz force in EM).
Perform a parity Eötvös experiment opposing enantiomorphic space groups P3(1)21 and P3(2)21 solid single crystal quartz test masses (opposite parity spatial distribution of atoms). Cultured quartz is a commercial product with exacting specifications. It is the only place nobody has looked. It is supported by orthodox classical gravitation theory that wholly includes GR as as restricted case. That case may be demonstrated incomplete without contradiction of prior observations.
Neil B. check out this thread for bloggy equation goodness:
Succumbing to Latex
Mike M, Scott H., Eastsider: I am well aware of the examples you give, that is why I carefully said “direct” measurement of Lorentz contraction. I meant direct in the strict conventional sense, of finding the length shorter by actual point to point comparison and not by any kind of inference, however convincing (and some are, but that isn’t the point when you refer to the strict definition of “direct”. I already said that LC is needed for consistency with time dilation, but that isn’t “direct,” nor is the inference that it only makes sense to imagine magnetic forces if we image LC of the line of charges, etc, sure. As for the MM experiment, LC is a theory to explain the null results: good and perhaps necessary, yes, but not “direct” in logical terms. It does not “prove” the LC, because there would also be a null MM result if for example “ballistic” (Galilean-style velocity addition) applied to light. Folks – you need to tighten up on the strictness of your semantics!
andy S: tx for the LaTeX info.
The magnetic field of current carrying conductor can (and should) be understood as a purely relativistic effect — you can derive it from electrostatics and the Lorentz transformations. You “observe” a length contraction every time you use an electric motor, and the (average) velocities involved are all tiny 🙂
Neil, you do have a rather strict notion of direct! I don’t think we’re going to have Lorentz contraction qualia anytime soon, and there’s no “logical” notion of directness for inductive reasoning. E&M is good enough for me, indeed it’s hard to see how you could make a “point to point” measurement of Lorentz contraction because it would come down to the famous “pole-barn” thought experiment which famously turns into a question of simultaneity — but you’ve rejected time dilation as sufficient to establish length contraction.
In general, you need a set of background assumptions to do any kind of science; you can’t establish Lorentz contraction without holding, and inferring from, some other beliefs about the nature of the physical world. Logically speaking, every measurement we’ve made is compatible with the absence of Lorentz contraction, but the amount of background you’d have to alter at this point is huge.
Simon, et al, the acceptance of simultaneity is indeed required in order to define “length”, but then once so accepted, the “length” has to be “directly” measured by finding endpoints at simultaneous times. (Well, I accept one other definition that is presumably equivalent: once we accept a “velocity” of a body, we can use how long it takes to pass a given point – but even then we need to define “velocity”. Actually that doesn’t require simultaneity if we can segue to rotary motion: you could use a large wheel of known circumference and take the period! This has interesting implications not often discussed anymore. Remember the old “Ehrenfest paradox”, and the related issues of elastic strain in a LCed rotating rim?)
That is what “direct” means, it is not too strict; you folks are too post-modernistically mushy etc. IMHO. If you use background assumptions and reasoning to infer something, well, it is still “inferred”, good practice or not. I’m not saying it’s wrong to believe in what is well inferred (I believe in LC too for all the same reasons), but I care about good use of wording – shouldn’t you?
BTW, what do any of you think of the proposition I put upthread, that we should use “mass” in the sense of “relativistic mass”? See my arguments for why it is better than the invariant “proper mass” as a placeholder for “mass” in general.
But Neil, whatever “point to point” measurement you make will require some background assumptions — if only in the physics of your measurement system! There’s really no way you can get a “direct” measurement.
If you are not willing to take the equivalence of frames as a given (from which time dilation gives you length contraction), then you are so crazy as to let anything go. You could measure length contraction but then say, oh, motion creates a mystical field that affects my measurement devices in a systematic way.
You suggest using a spinning disk to get around questions of simultaneity. The resolution of that paradox is far more “background loaded” than barn-pole, but let’s ignore that. I’ll do the experiment, but instead of painting dots on the sphere and watching them go by (your example of a “direct” experiment, I think), I’m going to charge it up and measure the magnetic field. There’s really no way you can separate those two into distinct epistemological categories (unless you have some wacky notion of qualia as ontologically prior — but I doubt you’re a Heideggerian!).
They are on a continuum and no “good use of wording” will make the difference discrete. Indeed, dots-on-a-sphere may require a great deal more background assumptions — about the nature of light propagation, for example — and the visual experience will have to be disentangled because of doppler shifts and so forth! I’m not particularly “post-modern” (whatever that means), but neither am I a logical positivist. This is pretty standard Quine stuff.
As for terminology questions of which mass is the “true” mass. In teaching relativity I find it’s nice to make a distinction between “number of atoms” (proportional to rest mass) and “mass”, the latter of which I define through inertia. Relativity does force us to complexify our intuitive ideas about “mass as stuff”, though, so perhaps we should bite the bullet and call them “zoog” and “zaag”.
Perhaps we should. Perhaps we don’t like being patronized. Or perhaps we are smart enough to recognize that it isn’t that simple. Why is using a ruler the “direct way” to measure a length? Isn’t counting the clicks on a click wheel a direct way to measure a length? Or perhaps sonar, which you have already accepted as a direct measurement, since it involves measuring the time it takes something (in this case a sound wave) to travel at fixed velocity from one end to the other. Why not time a light wave? Or you could lay a bunch of cigarettes of known length end to end and count them, which, I hope you agree, is really just a rather foolish kind of ruler. In fact, you could use any standard rod you like for the measurement: pingpong ball, cricket stumps, you name it. They are all rulers. Or even wavelengths of light of known frequency: you could set up a standing light wave and count the nodes. That would be an interferometer, which, I am sure you now agree, is just a ruler. So perhaps Messrs Michelson and Morley were just using a ruler to make a direct measurement, and we weren’t the ones who hadn’t thought through what constitutes “good use of wording.”
Well gee whiz, I stirred up quite a bit of wrangle hear which I suppose is what science and the philosophy of is all about. What I meant was, no one has yet measured a moving meter (at rest) stick by comparing endpoints, to say “Hey, we got 0.9999995 meters of length and it’s going 0.01 c, so we confirmed Lorentz contraction by direct measurement.” All that discussion about the sort of things that should also count is OK, but that just hasn’t been done whatever you want to call it, or call the related ways to find out. Can we agree at least on that?
Oops make that 0.001 c.