# The Most Elegant Solution In All Of Physics

A day or two ago, I posted my nomination for the greatest mystery in all of physics: why is it that the "gravitational charge" (i.e. how strongly you couple to the gravitational field) is identically equal to your inertial mass (i.e. how strongly you resist being pushed around by any kind of force)?

Einstein's General Relativity is our modern theory of gravity, and it answers this question in an extremely satisfying and elegant manner. Specifically, gravity is not a force at all; it's the geometry of spacetime. All objects move through spacetime in as straight a line as they can; if they deviate from a straight line, it's simply because of the curvature of spacetime. Objects of different mass are moving through the same spacetime geometry, so they all will move in the same manner.

This, to me, is an amazingly simple and elegant solution to what seems to be a great conundrum. Yes, it's often convenient to talk about gravity as a force, but when we recognize it not as a force but just as the background of what's out there, the conundrum completely goes away. Quantum Mechanics is in many ways a more successful theory than GR, in that it has been much more widely tested, and its tests are more precise. But I find at least the "gravity is the curvature of spacetime" part of GR to be far more elegant and beautiful than quantum mechanics.

The first glimmerings of gravity as geometry come from Galileo. Apocryphally, he dropped objects of a different mass off of the tower in Pisa, and observed them falling at the same rate. In reality, he did his experiments by rolling things down inclined planes, but the conclusion was the same.

Many of us intuitively expect heavier objects to fall faster. After all, they are heavier, so gravity is pulling on them more. Wouldn't they then fall faster? But, because they are heavier, it takes greater force to make them fall, and the two effects exactly balance out.

Our intuition serves us, however. Drop a hammer and a piece of paper-- and the hammer will hit the ground first. The problem here is that the two aren't operating just under the influence of gravity, but also under the influence of air resistance, and the air resistance has a much bigger effect (relatively speaking) on the piece of paper than it does on the hammer. But if we were to do this experiment on the moon— as, indeed, astronauts did— we would see the two objects fall at the same rate. On the moon, there is no atmosphere to speak of, so the two objects are indeed both under the influence only of gravity.

Construct the following situation: you have a big charged ball (you could think of it as a proton, but let's go bigger to avoid quantum effects). This big ball (of mass M) has a positive charge. Now start with two negatively charged balls, both with exactly the same negative charge. However, one negatively charged ball (m1) is heavier than the other (m2). Set the two moving so that they'll orbit the positively charged ball. The electrostatic attraction between them will hold them in orbit around the positively charged ball. However, that orbit will not be the same. If the heavier ball (m1) is in a circular orbit, and the lighter ball starts with exactly the same velocity, it will go into a larger elliptical orbit.

Gravity is different. Two objects the same distance from the Earth, starting with the same velocity, will orbit at exactly the same rate, regardless of their masses. Rather than treating gravity as a force, we say that in fact spacetime is curved around the Earth. The orbiting object wants to fly off in a straight line, but can't becuase of this curvature of spacetime. As a result, it just falls around the Earth. But the slope of the curvature of spacetime is the same for all objects, because it's the same spacetime— so two different objects starting with the same velocity have exactly the same orbits.

There is often an analogy used to a bowling ball sitting on a trampoline, but I'm not completely happy with that analogy-- because there is a "down" direction in that analogy that doesn't really have an analog in curved spacetime.

Gravity as the curvature of spacetime— it's such a simple, elegant, beautiful concept that it almost pains me to think that efforts to unify gravity with quantum mechanics may result in our learning that General Relativity is just the effective limit of a deeper theory (much as Newton's gravity is an effective limit of GR).

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I agree, what beauty. I don't think any other theory in physics can match it.

Yes QM has some higher precision tests--(what is the current best? Is it g-2?) but GR is no slouch--don't binary neutron stars test GR to something like 10^-9?

Honestly, I'm not sure the exact precision of anything; I think that the mag. moment of the electron is the most precise test of QM, but I could be wrong. But QM has been tested in a wider range of situations than GR has. And, GR hasn't really been seriously tested in the "strong field" regime in any serious way, so....

-Rob

I always wondered why if gravity was carried by a graviton field, how gravity could escape an event horizon. Now I'd like to know how we know for sure that distorted space returns to its original shape after a mass moves through that area. Dark energy anyone?

I always wondered why if gravity was carried by a graviton field, how gravity could escape an event horizon.

The question is probably a tautology, since event horizons are the result of gravity....

In any event, the curvature of spacetime is there outside the event horizon even if the mass is entirely inside it. The Schwarzschild Metric is a vacuum solution of Einstein's field equations. In quantum gravity, gravitatons are the quantized excitations of the metric, so it all works itself out somehow. Presumably.

Re: distorted space returning to its original shape after a mass moves through : if it didn't, then lots of measurements would be different. Gravitational microlensing events wouldn't be symmetric on the way up and the way down, orbits of moons around planets wouldn't stay regular, etc.

-Rob

So in your first post, you have the first equation of F = ma, and then second F = (GM/r^2)m, can the GM/r^2 be considered an acceleration? I'm a little rusty on my relativity theory, but wasn't one of the things that came out of general relativity was that an accelerated frame of reference (curvilinear motion?) is the same as being in a gravitational "field"... given that, it kinda makes sense to me that gravitational mass and inertial mass are the same!

I think a greater mystery, not yet solved, is why any mass imparts a curvature to spacetime. The fundamental particles with mass must do this, but at the QM level the effects are masked by the overwhelming relative magnitudes of the strong, weak and electromagnetic forces. But what is it about the mass that causes the curvature? Gravitons are excitations of the metric, but what exactly does that mean? How is it related to mass? Why does it result in a curvature of spacetime? Are there some particles with mass which might not cause curvature? For example, an electron is a very different beast from proton and neutrons. Or what about neutrinos, assuming they have mass, as somse dark matter theories propose? Is there something about quarks that might contribute to the curvature? Do particles without quark structure cause curvature?

By Wayne McCoy (not verified) on 27 Feb 2007 #permalink

Great post!
But I still feel gravity is the least understood phenomenon in the universe, in spite of all the successes of GR
Are the Pioneer anomaly and Maurice Allais' observations of the movements of a Foucault pendulum during a solar eclipse at all taken seriously by the community of physicists, or...?

By Johan Couder (not verified) on 27 Feb 2007 #permalink

but wasn't one of the things that came out of general relativity was that an accelerated frame of reference (curvilinear motion?) is the same as being in a gravitational "field"... given that, it kinda makes sense to me that gravitational mass and inertial mass are the same!

Brett -- yes. That's the equivalence principle. In a sense, everything about gravity being geometry follows from that principle-- so your instinct is right, everything about the two kinds of mass being the same does make entire sense in the light of that principle.

I just don't think that that principle is a priori obvious :)

I think a greater mystery, not yet solved, is why any mass imparts a curvature to spacetime.

Wayne -- yeah, I agree with that, and indeed that's the less elegant part of Einstein's theory in my view. (I've heard others make similar remarks.) If you notice, I did stick in a few disclaimers to stay in the regime of "small masses orbiting big masses," where you don't have to worry about how much the masses themselves are curving spacetime.

Why does the presence of energy density and pressure curve spacetime? Dunno. But if we postulate that it does, and also the equivalence principle (i.e. that things just move as straight as they can through that spacetime), we get the beautiful theory of GR out of it. And, the key point is, the predictions of GR match very well with experiment, so GR seems not only to be beautiful, but to be right to the precision with which we've tested it to date.

I have been meaning to post this question in several places, but this seems to be the best candidate.

I understand that GR posits that gravity is not a force, but that it is a curvature induced by massive bodies in spacetime.

I also understand the background of GR, that test particles move along geodesics, and that geodesics are the curvature of spacetime induced by massive bodies. I haven't mastered the mathematics of it all, but that's the gist of it.

That leaves me with the following question. Consider a universe consisting of two bodies, one a massive body that induced the curvature of spacetime, and the test particle, which also has some mass (i.e., not a photon, which is required to move along null lines), that is in the geometry induced by the massive body, but is of insufficient mass to induce much of a curvature in its immediate vicinity. Further consider that the test particle is not moving with respect to the massive body.

What GR's theory that gravity is not a force, but induces a curvature such that particles moving along a geodesic, suggests to me is that, if the test particle is not moving with respect to the massive body, it will not be attracted to the massive body because it is not moving. Is that correct? If so, I find that very difficult to believe, although I do appreciate that it is possible.

Galileo's apocryphal Pisa experiment doesn't really apply as a counter-example, because, obviously, since the planet is rotating, the feather and the cannonball would be moving with respect to the earth, and, hence, after having been dropped, they would be moving along a geodesic, and they were moving even while being held by Galileo.

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An example case to illustrate E = MC2 in several textbooks (see Feynman), is of a gas in a container being heated. Now I can see as the velocities of the molecules increase, time slows down. So therefore the inertia of the container appears to increase. But why should the weight (gravity effect) of the container change? (I have been struggling with this question for a while.)

By Larry Brooks (not verified) on 28 Feb 2007 #permalink

raj: The important thing to realize is that objects move along geodesics in the 4D spacetime, not in 3D space. In a flat spacetime, the shortest distance between here-and-now and someplace-and-now+dt is for someplace=here, ie. no motion. But in a curved spacetime, that's no longer true: here-and-now is closer to here+dx-and-now+dt than it is to here-and-now+dt, so you get motion.

I don't even begin to understand all this, but I do find it fascinating. However, I was watching an astronomy show a couple years ago, on the Science channel, I think. They might have been talking about science vs sci-fi, and warp engines and all that. I remember the astronomer saying something to the effect that while the theory of warping space or taking advantage curved space is intriguing, our best evidence shows that spacetime is flat, not curved at all.

Jer | February 28, 2007 08:10 AM

Oh, that's right (slaps head). I had forgotten about that aspect. I was considering geodesics in 3d space, not 4d spacetime. Thanks very much. That clears it up for me.

Jeff: spacetime being "flat" refers to the very largest scales. Imagine a bowling ball. On the large scale, the surface is curved in one way: it is a sphere. But on the small scale, there are pits and scrapes on the surface, so the curvature is different at almost every point. The flat spacetime comment means that space is more like a sheet of paper than a bowling ball: flat at very large scales instead of curved at very large scales, ignoring local tiny curvature due to pits and scrapes.

By Ross Presser (not verified) on 28 Feb 2007 #permalink

Great post. I hope this question is not too far off topic.
I have seen it mentioned that the speed of propogation of gravity waves is substantially greater than the speed of light. I have also seen that Einstein assumed in GR that it was equal to the speed of light? What do recent experiments conclude on the matter. Do we have a reliable figure yet?

By Michael H (not verified) on 28 Feb 2007 #permalink

Great post, its just so breathtakingly elegant!

By Ahsan Ali (not verified) on 28 Feb 2007 #permalink

I believe GR also says that mass CREATES space-time. One could imagine that mass creates space sends it out like light from a candle. The closer you get to the mass, the greater the density of the space. Therefore space is not curved near mass, thereï¿½s just more of it. Other mass moves or ï¿½fallsï¿½ into the greater quantity of space. I envisioned this similar to buoyancy but that may be a stretch.

I am not a scientist, but I do apreciate the elegance of a well-stated idea.
I've encountered similar descriptions of GR, but it has never really sunk in. I think my brain must be stuck in the "intuitive version" of what gravity is. I suspect most people think of it as a pulling force, at least until they begin to study the related sciences.
So, please, tell me, do concepts such as these become easier to understand as one mulls them over? Is it better to just accept, and hope for clarity later?

I have seen it mentioned that the speed of propogation of gravity waves is substantially greater than the speed of light. I have also seen that Einstein assumed in GR that it was equal to the speed of light? What do recent experiments conclude on the matter. Do we have a reliable figure yet?

Well, if you start with Einstein's field equations, you can end up with a wave equation showing gravitational waves propagating at the speed of light. (Pedantic aside: they aren't "gravity waves". "Gravity waves" are waves driven by gravity, e.g. waves on the surface of water. Waves in the fabric of spacetime are called "gravitational waves".)

Since gravitational waves haven't been detected directly, I suspect there's no direct proof that they move at the speed of gravity. However, gravitational radiation has been detected indirectly (in the slowing of a binary pulsar), and there was an experiment a few years ago showing that the gravitational effect of Jupiter (if memory serves) does in fact move at the speed of light... yeah, here's a NRAO (National Radio Astronomical Observatory) press release on it: http://www.nrao.edu/pr/2003/gravity/.

John -- well, it's always best to try to achieve clarity. Accepting is good if you really trust the people whose assertions you accept, of course, but understanding is better!

As for "getting it" -- it's tough. I'm not sure if I get it better than I did in the past, or if I just got used to the idea.

I'm teaching a GR class right now, and we were dealing with some of the descriptions you use of spacetime around a black hole. One description is called the "Schwarzschild Metric," and seems intuitive in that two of the variables are "r" (and it is exactly distance along a radial direction for a very distant observer) and "t" (which is exactly time for a distant observer). Another is the Kruskal-Szerkes metric, which uses seemingly more abstract variables U and V. Students were saying it was very difficult to get any physical intuition for the latter. I started by trying to destroy their false intuition in the r and t of the Schwarzschild metric... they don't really mean what you think as you get very close to (or, especially, cross) the event horizon of the black hole.

Then we step back and start asking more concrete questions. OK, which way does light go if we use these variables? What are the results of some experiments or questions we could ask? How would it be different if we didn't have curved spacetime? Etc.

One way I link to think about curvature is by measuring the interior angles of a triangle. Do the following: start at the Equator of the Earth, and walk due north. Walk in as straight a line as you can until you hit the North pole. then make a 90 degree turn to the right, and head South back to the equator. Make a 90 degree turn to the right, and head along the equator until you hit your starting point.

You've just drawn a triangle on the 2d space that is the surface of the Earth. Yet, the three interior angles of this triangle do not add up to 180 degrees as we expect from "normal" geometry, they add up to 270 degrees!

-Rob

I would like to clarify how my previous question relates to the central issue of this post. Under Special Relativity there appears to be a way to increase inertial mass without out increasing the force of gravity, a violation that gravitational mass and inertial mass are equal: That is that you heat up a container of gas. Since mass is invariant the gravitional force of the container should not change. Yet because of time dilation via the increased velocitity of the molecules within the container, the container appears to have more inertia. Now I am asking how under Special Relativity this apparent paradox is resolved? What was the viewpoint before General Theory? I realize that I may be making a dumb mistake in the phrasing of the question, but if so, I would greatly appreciate some illumination. I also assume that there was a consistent way of looking at this under Special Relativity.

By Larry Brooks (not verified) on 28 Feb 2007 #permalink

and there was an experiment a few years ago showing that the gravitational effect of Jupiter (if memory serves) does in fact move at the speed of light...

Rob, that interpretation was rather controversial. In my view, the measurement didn't establish that "gravity propagates at the speed of light". Rather, what it established was that the metric of a pointlike source transformed under Lorentz transformations just as it should. In other words, the "electric-like" gravitational effect of Jupiter's mass in its own rest frame produced a "magnetic-like" gravitational effect due to its motion as viewed in a different rest frame. The effect is analogous to the magnetic field produced by a moving point charge.

What was amazing about the experiment, from an observational point of view, is that the "anomolous" lensing effect due to this magnetic-type effect is smaller in magnitude from the electric-type lensing by a factor of v_{orb}/c, where v_{orb} is Jupiter's orbital velocity (as seen from the observer's frame ... i.e., Earth). And yet they got it! It's a real pity that this amazing measurement was somewhat obscured by the brouhaha over its interpretation.

Hi Larry:

Since mass is invariant the gravitional force of the container should not change.

The rest mass is invariant. When you've heated it up, it ain't the "rest mass" that you're looking at. Or, to put it another way, if you just view the box of gas as a closed system, you've gone from one rest mass to another (cold rest mass to hot rest mass). You've changed the system.

Since mass is invariant the gravitional force of the container should not change. Yet because of time dilation via the increased velocitity of the molecules within the container, the container appears to have more inertia. Now I am asking how under Special Relativity this apparent paradox is resolved? What was the viewpoint before General Theory?

First, special relativity says nothing about the gravitational force of the container. It can't: Gravity and special relativity are incompatible. So, all special relativity can say is "Make the contents of the box hot and its inertia is increased" (as in the discussion above).

General relativity inherits all of special relativity, so in GR this viewpoint from SR comes along for the ride. If we interpret this in Newtonian language, both the "inertial mass" and the "gravitational mass" change in lockstep with one another. In relativistic language, the box follows a geodesic of spacetime, independent of whether it's hot or cold, just as Rob has described.

Hi Scott;

- There is only one mass, what is sometimes called "rest mass", is simply "mass." Yes, that mass gets divided by an extra factor in the mechanics of moving bodies under Relativity. But the term m is invariant according to all I have read. The additional effects when heated are due to length contraction and time dilation, not a change in "m."

- E=MC2 does say something about gravity, for it says that a system of high internal energy weighs more than a system of low internal energy. Einstein specifically says this at the end of his 1905 paper.

Hope we can shed more light on this.

By Larry Brooks (not verified) on 28 Feb 2007 #permalink

The additional effects when heated are due to length contraction and time dilation, not a change in "m."

No, it's exactly due to a change in "m". Length contraction and time dilation have no impact on this discussion.

Consider a box full of cold gas. Consider a single atom in this gas. Cold means that the kinetic energy of that atom --- computed using its velocity relative to the rest frame defined by its box --- is small. Let's say it's so cold that, for all practical purposes, the kinetic energy is zero. Now, imagine all the atoms in your box have this kinetic energy. Then, the "rest mass" of the box's contents is just the sum of the rest masses of all the atoms in the box. Call that m1.

Now, take a second identical box, and fill it with hot gas. The kinetic energy of a typical atom in this box is very large. This means that the typical energy of an atom in the box is higher (perhaps considerably higher) than its rest energy (rest mass times c squared). To get the total energy of the atoms in the gas, I sum all their rest masses, multiply by c squared, then add up all their kinetic energies.

All these hot atoms are trapped in the box, so the details of them zipping around are lost to us. All we know is that the gas in the box has some "apparent" rest energy which is given by adding up (m_{rest} c^2) + kinetic energy)/c^2 for all the atoms. We can determine this by applying a calibrated force and measuring the accleration. Call that m2. Special relativity tells us m2 > m1: Because the atoms of the hot gas have kinetic energy, and because of the equivalence of mass and energy, the box of hot gas has greater inertia than the box of cold gas.

E=MC2 does say something about gravity, for it says that a system of high internal energy weighs more than a system of low internal energy.

Fair enough. What this is really saying though is that Einstein assumed in 1905 that gravitational mass and inertial mass remained the same. He didn't really have a strong justification of this until after he formulated general relativity, though.

Just occurred to me that, in my response to Larry, my answer kind of lost sight of the point I was trying to make. Larry's last point really gets to what I'm trying to say: The hot system has lots of internal energy. However, if both the hot and the cold systems are closed up in some way (say inside a really nice Thermos, so I can't tell which is cold and which is hot just by touching), then I don't know which one has lots of internal energy and which does not. All I can do is measure the inertial mass, which I will interpret as the "rest mass" of the system.

My point being that the notion of "rest mass" can be a little more complicated than discussions often make clear. It's really only a clean, invariant quantity for things that are really simple --- e.g., a single electron. If you have a composite object with lots of possible interactions, "internal energy" can be measured and interpreted as rest mass. For such a system, I can change the rest mass of the composite object by, for example, making it hot.

None of which invalidates the whole point of Rob's post: The "gravitational mass" and the "inertial mass" are identical whether the object is hot or cold, and GR provides an excellent way of explaining why. (Sheesh ... Rob, sorry for making this so complicated!)

I think we are starting to get on the same wavelength. I am not disputing Rob's point, I am simply trying to understand SR better. Your 2 nd paragraph is close to what Einstein says in 1905. However, you can't say it has nothing to do with length contraction and time dilation because that is exactly the SR effects that Einstein used to come up with E=MC2. My goal is to understand better how he connected these concepts. (That is why I have posted this question.) The two pages where he connected mass and energy were written 3 months after the first part of the paper and almost didn't make it to the publisher on time. So Einstein gave it a lot of thought. There seems to be an implication that SR does not lead to E=MC2, only GR....In the specific case of a container of gas time dilation will happen as the molecular velocities increase whether you think it is relevant or not. So it has to be taken into account.

By Larry Brooks (not verified) on 01 Mar 2007 #permalink

Thanks for the clarification Ross!!

a great post, and it's fun to discover a new astronomical blog. welcome to Sb!

yet another example of einstein's unparalleled genius as observed by a layperson (mself) with only a physics II background :)

Now I am asking how under Special Relativity this apparent paradox is resolved? What was the viewpoint before General Theory?

What Scott said. A container of gas heated up so that it has more energy, when viewed as a closed container, will have higher m, just as a box with two hydrogen atoms and one oxygen atom will have a higher mass than a box with one water molecule.

As regards General Relativity : it handles all of this by not making mass per se the source of the curvature, but the stress energy tensor. That includes mass, kinetic energy density, pressure, and transverse mometum flux all in one big rank-2 geometrical object. (Pressure just being longitudinal momentum flux, by the way.) None of those things individually are a relativistic invariant (except for rest mass)-- but the whole tensorial thing is.

In re-reading all the responses I think there is a lot that makes sense. But I want to focus on what doesn't.

- Scott said the increase in weight had nothing to do with time dilation. That can't be true. Otherwise there would be a difference between the inertial and gravitational mass.

- Scott said there is a difference between a container with one molecule versus many molecules. Molecules in a gas are very far apart and don't have strong forces between them. Time dilation has to play the same role in both situations. So if the velocity increases for one molecule it has to have a similar effect for many molecules...........Rob your statement is different than Scotts. There are strong forces between hydrogen atoms and an oxygen atoms.

- Rob, if mass invariant, as you agree, then your first sentence doesn't make sense to me. Are you saying that gravitational force is not proportional to m, but to m/beta? (Where beta is the well know Relativity coefficient.)

- Scott and Rob, there was an implication that E=MC2 can only be understood through GR. Do you feel Einstein made a mistake in the 1905 paper?

- Unfortunately although I have a degree in physics, I don't understand tensors yet, so your second paragraph doesn't help me much. I do plan to eventually try to understand GR, but I want to understand SR first. This seems a logical progression to me.

Larry

By Larry Brooks (not verified) on 02 Mar 2007 #permalink

Larry Brooks | March 2, 2007 08:19 AM

It may be a bit of overload, but you might want to take a look at Sean Carroll's GR notes available through http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll_contents… I downloaded and am going through the PDF version available through http://arxiv.org/pdf/gr-qc/9712019 He goes through a lot of tensor algebra.

The problem that I have with modern physics (my degrees were in the early 1970s) is that physics seems to have become little more than a branch of mathematics (viz. string theory), whereas earlier many mathematical tricks stemmed from developments by physicists. In other words, the mathematics tail seems to be wagging the physics dog.

RAJ,
Thanks for that tip on what looks like a wonderful resource. Surprising its free considering that many physics textbooks often sell for hundreds of dollars. I also got "The Road to Reality" by Penrose out of our local library. Its amazing that he claims to have written it for the "layman."

My degrees go back a long way also. I am now retired and have the time to try to get a better understanding of physics.

By Larry Brooks (not verified) on 02 Mar 2007 #permalink

Hi Larry --- just fyi, I'm planning to answer in more depth, but I've been swamped and had no time. (This is the first blogging I've looked at in two days; and I'm doing it as a form of procrastination while the caffeine drips into my brain.)

One thing that confuses me is why you think I've implied that E = mc^2 comes from GR. It *definitely* doesn't --- it's totally SR. I also don't understand why you think it comes from time dilation; that's very confusing to me.

I suppose you could argue that both time dilation and E = mc^2 are consequences of Lorentz transformations, and in that sense have a common origin. Is that what you're saying?

Scott,
I look forward to little more discussion on this....E=MC2 seems to invoke gravity because in several papers Einstein says that radioactive processes might demonstrate a weight change due to changes in internal energy...as perhaps turned out to be the understatement of the century. Yet you have said (and maybe convinced me)that SR says nothing about gravity. Yet here Einstein is clearly saying the proof of SR will be by measuring gravitational forces.

The other point of confusion involves whether you are talking about mass, m, or what is sometimes called relativistic mass, m/beta. When I was in school a major point was made that m was invariant and it only led to confusion to talk about relivitistic mass. And indeed,it seems easier to view inertia as an effect of time dilation than a changing mass. However, if interia and gravity change together, as was suggested, then m/beta is indistinguishable from a change in mass. So I am caught between these two different viewpoints.

In the case of the container of gas, it seems to me after reading all the posts, that time dilation must cause the increase in inertia and the increase in weight.

By Larry Brooks (not verified) on 02 Mar 2007 #permalink

Larry Brooks | March 2, 2007 10:50 AM

No. E=mc^2 definitely comes from SR, not GR. Indeed, in GR, there is no conception of conservation of energy (hence mass) globally--only locally. My SR text (from the early 1970s) describes the equations leading up to E=mc^2, but I can't find a description online. You might try clicking through http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html for an explanation (no guarantee, but it's a decent site.)

Going up a bit, regarding a comment by Larry...

Scott said the increase in weight had nothing to do with time dilation. That can't be true. Otherwise there would be a difference between the inertial and gravitational mass.

Larry, you have to distinguish between weight and mass. Weight is the application of gravity to a massive object (i.e., an object that has mass, not a big, heavy object). Once you get the lingo straight, it all falls into place.

Per the last sentence of the previous post, one other possible way of viewing this under SR, is that length contraction distorts the gravitational field so that the force inceases while time dilation increases inertia in such a way that two effects are equivalent to an increased mass.

By Larry Brooks (not verified) on 02 Mar 2007 #permalink

Relativistic mass is passee. Nowadays, when we say "mass", we mean "rest mass". There's not really such a thing as relativistic mass.

The *energy* of a particle of mass m, as measured by an observer moving at velocity v with respect to that particle, is E=mc^2/sqrt(1-v^2/c^2). m is just mass (sometimes called "rest mass," but that term is redundant nowadays), and is invariant.

So, let me slightly correct what I said above. Yes, m is invariant, but it's all energy density that is the source of gravity in GR, and energy density is not invariant. (Just consider length contraction; energy/volume is gonna change because of that, but also energy changes because stuff will all be moving at different speeds.) The stress/energy tensor *is* an invariant, whereas just raw energy density isn't. The S/E tensor encapsulates energy density, pressure, and transverse momentum flux. So, in Einstein's field equations, the Stress-Energy tensor is the source term that gives rise to spacetime curvature.

-Rob

- In my post of 11:10 I was refering to the last sentence of my post of 10:50 AM, not the previous post as E-mails may have crossed in the "aether."

- RAJ, I am not clear on what the meaning of your comment regarding the difference between mass and weight is. Could you expand a little on that?

By Larry Brooks (not verified) on 02 Mar 2007 #permalink

Scott,
Some second thoughts:

- Its possible your first simple explaination is correct and answers my question, "What this is really saying though is that Einstein assumed in 1905 that gravitational mass and inertial mass remained the same. He didn't really have a strong justification of this until after he formulated general relativity, though."

- In the case of nuclear energy or just a compressed spring there appears to be no SR mechanism, length contraction or time dilation, that comes into play. So maybe I am wrong to look for such.

- It is pretty amazing if this is the case that he arrived at a relationship way beyond SR, in a sense.

- Most of the derivations in popular books of E=MC2 don't satisfy me. But I am now reading the explanation in Pais's book which seems much more substantial. I will spend some time on that.

By Larry Brooks (not verified) on 02 Mar 2007 #permalink

Hi Larry ---

End of the day, and procrastinating on my lecture notes for next week ... a great time to catch up on this blog conversation ;)

From your last post, it sounds like our wavelengths are starting to sync up now! Let me just hit on one point, which is this connection between E = mc^2 and time dilation which keeps coming up, and which is a source of some confusion. Please forgive the somewhat pedantic tone of this email; I want to go back to the beginning, so this will be elementary. The Pais book will surely be more useful for the details; my goal here is to illustrate conceptually how time dilation and E = mc^2 arise from the concepts of special relativity.

Special relativity was originally formulated because electrodynamics was inconsistent with the mechanics of the day. Mechanics taught us that the speed with which a wave was measured would vary with the observer's speed. E.g., standing next to a river, I see a wave propagating down the river, and measure its speed as say 5 meters/second. If I run quickly enough, I can keep up with that wave; in my "running reference frame", the speed of the wave is zero.

Maxwell's equations showed us that in ALL reference frames the speed of light was c. Light is apparently very different from other kinds of waves! Einstein noted that electrodynamics was extremely successful, and that, despite this weirdness, it was unlikely that Maxwell's theory required modification. So, he decided to take "all observers observe the speed of light to be c" as the initial ansatz and see what happens.

As we know from history, requiring c to be the same to everyone means that different inertial observers have different ideas about what space and time are. As I run along the river, my natural time coordinate is a mixture of what I called time and space when I was sitting still. Time dilation is a consequence of this mixing of time and space: As I run past people, they notice that my watch is running slow. I, of course, claim that their watches are running slow, since I define my frame as at rest and claim that they are moving. And we're both right.

What happens to the laws of mechanics once space and time are mixed up? Einstein wanted there to still be laws of conservation of energy and conservation of momentum. However, with space and time all screwed up like he made them, some of the details had to be changed. In particular, he found that "energy" and "momentum" got mixed together in much the same way that "space" and "time" got mixed together. And, the conservation laws only worked out if he assigned an "energy" to masses that are at rest (with respect to some frame). That energy is of course E = mc^2.

Today, what we say is that space and time need to be combined into "spacetime" (a notion that was introduced by Minkowski, and that Einstein initially sneered at). Different observers have different ways of splitting spacetime into "space" and "time". Time dilation is a consequence of the different splits of different observers. Likewise, we really need to combine "energy" and "momentum" into a unified object, and recognize that different observers split it into different things. To get this unification to work in a way that preserves conservation laws, and gives us the proper "low velocity" physics, we need to assign all masses a "rest energy" of mc^2.

I think this gets to the heart of why I was confused by your assertion that the increased energy was due to time dilation. Both affects arise from the basic physics of special relativity, particularly the manner in which quantities must be transformed as we bop between reference frames. However, they are different aspects of that basic physics. Saying that one was due to the other confused me.

Wow, this turned out to be WAY longer than I expected. Sorry!

General

- I think its wonderful that you guys donate your time to answer questions from such a diverse group of people that you don't even know. I plan to check your site out on a regular basis. Hope you keep it up for a long while.

- On the mechanics of how the blog works: I see you do archive stuff. Is it possible to go back and ask additional questions on an old thread? Do you come up with new subjects (threads) on a periodic basis or do you respond to reader questions?

- On this specific thread, my focus on time dilation was not that one thing causes another. My thought was that in the case of a container of gas, time dilation will occur as the molecules speed up (correct?), therefore inertia will increase, therefore if we assume gravity and inertia must go together, the weight of the container should increase. So maybe in this one specific example there is a mechanism for E=MC2...But I admit I have to read more before I go around more in circles.

By Larrry Brooks (not verified) on 03 Mar 2007 #permalink

Larry Brooks | March 2, 2007 11:20 AM

RAJ, I am not clear on what the meaning of your comment regarding the difference between mass and weight is. Could you expand a little on that?

Short answer, not really. Long answer, as far as I can tell, mass is a value that reflects primarily the momentum (F=dp/dt) of a body, whereas weight is a value that reflects the effect of the gravitational force of that body. It's a subtle distinction, but a real one. A body has the same mass on the earth as it does on the moon, but the weight on the earth and the moon are very different--by a factor of about six, because of the difference in gravitational attraction between the earth and the moon.

Scott H. | March 2, 2007 06:24 PM

Much of this is correct, but I'll take issue with

Maxwell's equations showed us that in ALL reference frames the speed of light was c.

As far as I can tell, it wasn't Maxwell's equations that showed that, it was the Michaelson-Morely experiment, which failed to discover a preferred reference frame (that of the ether), which would have been entirely consistent with Maxwell's equations.

It was the obvious inconsistency of the presumptions behind Maxwell's theory with observed reality that gave rise to SR. And, as far as I can tell, SR preserves Maxwell's equations pretty much intact.

Hi Raj --- actually, you're right. I should have been more careful. The invariance of the wave speed only really comes about when the transformation law is, in fact, the Lorentz transformation.

Oops.

Scott,
Do you think looking at the incease of weight of a container of gas as its heated as an effect of time dilation (as posted at 8:36 AM) might be mathematical identical (lead to the same result) as applying E=MC2 ?
Thanks,
Larry

By Larry Brooks (not verified) on 03 Mar 2007 #permalink

Rob Knop | February 28, 2007 03:39 PM

Rob, I hate to intrude yet again, but I have a question. I'm moderately familiar with the Schwarzschild metric, and with the r=2gm issue, and the K-S transformation that eliminates the coordinate system selection issue regarding the singularity at r=2gm.

The question that I have is as follows. Most texts suggest that even with the K-S tranformation, there is still a singularity issue at r=0. I'd acknowledge that, but I'm wondering whether the issue actually doesn't exist. Since m is a measure of the mass inside the radius r, and the mass should be expected to be a function of the volume at radius r (4/3 pi r^3), it seems to me that in the limit the mass would go to zero faster than the radius. So where is the singularity?

Great blog, by the way.

Hi Larry ---

Do you think looking at the incease of weight of a container of gas as its heated as an effect of time dilation (as posted at 8:36 AM) might be mathematical identical (lead to the same result) as applying E=MC2 ?

It's possible; I'm not quite certain how to do the calculation as you describe it, though. Ie, what aspect of the time dilation actually leads to the increase in weight? It would be complicated: it would need to be done molecule by molecule, then averaged using a statistical weighting function for all the distribution of molecules in the gas.

Hi Raj ---

Most texts suggest that even with the K-S tranformation, there is still a singularity issue at r=0. I'd acknowledge that, but I'm wondering whether the issue actually doesn't exist. Since m is a measure of the mass inside the radius r, and the mass should be expected to be a function of the volume at radius r (4/3 pi r^3), it seems to me that in the limit the mass would go to zero faster than the radius. So where is the singularity?

For a black hole, the mass distribution is truly singular (at least classically): it's confined to a point of radius 0. You can regard it as having a density that is a Diract delta function. So the mass does not go to zero faster than the radius.

This was first stated rigorously by Penrose; what I'm trying to get to here is an intuitive re-stating of the singularity theorem that he proved in the late 60s. Basically, he realized that the spacetime contained an outermost trapped surface (a surface in which bursts of light are not expanding), then interior to that surface bundles of light rays must be focused. If light rays are focused, then matter must be even more strongly focused, since matter can't travel as fast as light. From here, it is a simple matter to get all the matter piling up at a single point. So you approach (rather quickly, as it turns out) the limit of the Schwarzschild metric --- all matter squashed into a point singularity.

(Of course, this is purely classical physics ... quantum gravity presumably would have something to say about this!)

Scott,
I do envision it as being a complex calculation as you state. The rational is that time dilation is equivalent to inertia; as time moves more slowly, an object responds more slowly to a force. And the same factor divides time that divides momentum in the transformation.

"Ie, what aspect of the time dilation actually leads to the increase in weight?"

That was my original question !! In the more conventional viewpoint, why should gravity increase just because you divide momentum by a factor? Your answer before that Einstein assumed inertia and gravity went together is probably correct. And the same rational would appear valid for the time dilation viewpoint. It's curious that the old concept of "relativistic mass" seems the most intuitive in this case.

I am surprised that Einstein didn't mention this assumption. He was so careful in laying out very subtle assumptions in later talks on SR (per Pais.) From prior posts I get the impression that this issue is resolved by GR.

By Larry Brooks (not verified) on 04 Mar 2007 #permalink

Scott,
I see some confusion in my prior post and you may see more !

Time dilation is a kinematic effect described by the Lorentz transformation. The transformation of F=ma into a moving reference frame takes into account both the length contraction and time dilation of the Lorentz transformation...So I should have said, "the same factor that divides momentum in the transformed F=ma divides time in the Lorentz transformation." Feel free to criticize, I greatly appreciate your feedback.

By Larry Brooks (not verified) on 04 Mar 2007 #permalink

Scott H. | March 4, 2007 02:41 PM

Thanks for the information. I did some research over the Web on the Penrose singularity theorem (that was after my time at university) and it pretty much falls into place.

RAJ,

From Mar 2, 11:08 AM
"Larry, you have to distinguish between weight and mass. Weight is the application of gravity to a massive object (i.e., an object that has mass, not a big, heavy object). Once you get the lingo straight, it all falls into place."

That is exactly what I am trying to do. Time dilation is the root SR effect that leads to an increase in inertia, or momentum, or relativistic mass, depending on the way you interpret the factors. (There are only two root SR effects; length contraction and time dilation.) All ways of looking at the factors are mathematically equivalent. Yet with time dilation as the root cause, it does not appear obvious to me that weight goes along with inertia. It has to be an assumption.

I was just reading the Feyman lectures, chapter 15 and 16 on Relativity. The assumption that there is such a thing as "Relativistic mass" that exhibits both inertia and weight is stated both explicitly and implicitly numerous times.

By Larry Brooks (not verified) on 05 Mar 2007 #permalink

Rob,
I can put my question much simpler now:

Under SR, one has to either equate gravitational mass with m, the invariant inertial mass, or m/beta the "inertial relativistic mass." The correct choice is the latter. Under SR this is a subtle assumption. It implies that time dilation influences both inertia and gravity. Per your post this assumption is not needed in GR.

Would you agree with this summary?

By Larry Brooks (not verified) on 07 Mar 2007 #permalink

Hey all -- I've been AWOL for a few days, so I've fallen behind in the discussion. I'll catch up shortly, but I have some other posting to do first.

Rob and Scott,
This will be my last post on this thread unless there is more interest. I think only three statements are confusing and one was made by me:

Larry said:
'There is only one mass, what is sometimes called "rest mass", is simply "mass." Yes, that mass gets divided by an extra factor in the mechanics of moving bodies under Relativity. But the term m is invariant according to all I have read. The additional effects when heated are due to length contraction and time dilation, not a change in "m."

Scott said:
'No, it's exactly due to a change in "m". Length contraction and time dilation have no impact on this discussion."

Rob said:
'Relativistic mass is passee. Nowadays, when we say "mass", we mean "rest mass". There's not really such a thing as relativistic mass.'

My statement is wrong: When you put more energy in a system the mass increases. Invariant only refers to a snapshot of the mass in the lab frame under equilibrium conditions with regards to energy transfer, as opposed to a moving frame.

Scott's statment is somewhat confusing because time dilation is the root cause of the increase in inertia.

Rob's statement is the one I heard when I was in school. But as a result of this discussion, it seems to me that the concept of "relavitistic mass" is essential to understanding SR. In fact it seems less abstract than the concept of invariant mass. Invariant mass is the same nathematically as mass/beta but with velocity set to zero. Gravity has to be related to "relavitistic mass" because invraiant mass is just a snapshot, in a sense.

By Larry Brooks (not verified) on 11 Mar 2007 #permalink

Larry -- no. In General Relativity, "mass" is still the same as "rest mass".

Gravity is related to more than "relativistic mass." It's related to the whole stress-energy tensor, which incorporates the kinetic energy in a big and complicated way of all moving particles, as well as their mass (i.e. their rest mass). That takes care of what you're talking about (i.e. the energy of a particle being different as measured in different frames of reference).

-Rob

Rob,
In the case of a containter enclosing gas molecules, is it correct to say that the weight in the laboratory frame (where the contaner is stationary) is equal to the sum of the relativistic masses of all the gas molecules (where each molecules weight is affected by its velocity)? I think this is what Scott was trying to say in his first reply.
Larry

By Larry Brooks (not verified) on 11 Mar 2007 #permalink

Hi Larry --

Sorry I dropped out of this discussion; I've been overwhelmed with various duties lately and have not had time to follow any blogs for most of the past week.

Just to chime in: Your summary of stuff in this last posting (2:39 on March 11) is indeed pretty much what I was saying. Slightly more precisely, what I would say is that the sum of the relativistic masses of all the gas molecules is proportional to the weight (rather than equal to).

Rob: The orbiting object wants to fly off in a straight line, but can't becuase of this curvature of spacetime.

Sorry, this is a goof, and not just the typo. Your later discussion shows that you do understant the principles, but just for the record: The object "wants" to move "straight ahead", on a path that looks locally like a straight line. Due to the gravitational curvature, that path runs around the planet in an ellipse. (Actually, a 4-D elliptical helix winding around the planet's own world-line.)

Regarding the heated box of gas, the simplest way to look at it is this: You have a box of gas. If you add energy to it, the mass of your box will increase, exactly, by the mass-equivalent of the energy you added. If the energy's mass-equivalent is comparable (similar order of magnitude) to the gas' rest mass, then when you examine the gas, you will find the particles moving at relativistic speeds. (And your "gas" will have become a scary-hot brew of plasma/hard radiation, contained only by the Unobtanium walls of your problem box. ;-) )

By David Harmon (not verified) on 21 Mar 2007 #permalink