I love telling the stories of Neptune and Vulcan. Not the Roman gods, the planets that were originally hypothesized to explain the mysterious motions of other planets. Neptune was propsed by Urbain Le Verrier in order to account for deviations from the predicted orbit of Uranus. After it was discovered, he tried to repeat the trick, suggesting a new inner planet, Vulcan, to account for the deviations of the orbit of Mercury. It didn’t work the second time; Einstein’s general relativity, not a new celestial body, was the ultimate explanation.
In other words, Neptune was dark matter, and it was eventually discovered. But for Mercury, the correct explanation was modified gravity.
We’re faced with the same choices today, with galaxies and clusters playing the role of the Solar System. Except that the question has basically been answered, by observations such as the Bullet Cluster. If you modify gravity, it’s fairly straightforward (although harder than you might guess, if you’re careful about it) to change the strength of gravity as a function of distance. So you can mock up “dark matter” by imagining that gravity at very large distances is just a bit stronger than Newton (or Einstein) would have predicted — as long as the hypothetical dark matter is in the same place as the ordinary matter is.
But it’s enormously more difficult to invent a theory of modified gravity in which the direction of the gravitational force points toward some place other than where the ordinary matter is. So the way to rule out the modified-gravity hypothesis is to find a system in which the dark matter and ordinary matter are located in separate places. If you see a gravitational force pointing at something other than the ordinary matter, dark matter remains the only reasonable explanation.
And that’s precisely what the Bullet Cluster gives you. Dark matter that has been dynamically separated from the ordinary matter, and indeed you measure the gravitational force (using weak lensing) and find that it points toward the dark matter, not toward the ordinary matter. So, we had an interesting question — dark matter or modified gravity? — and now we know the answer: dark matter. You might also have modified gravity, but one’s interest begins to wane, and we move on to trying to figure out what the dark matter actually is.
But some people don’t want to give up. A recent paper by Brownstein and Moffat claims to fit the Bullet Cluster using modified gravity rather than dark matter. If that were right, and the theory were in some sense reasonable, it would be an interesting and newsworthy result. So, you might think, the job of any self-respecting cosmologist should be to work carefully through this paper (it’s full of equations) and figure out what’s going on. Right?
I’m not going to bother. The dark matter hypothesis provides a simple and elegant fit to the Bullet Cluster, and for that matter fits a huge variety of other data. That doesn’t mean that it’s been proven within metaphysical certainty; but it does mean that there is a tremendous presumption that it is on the right track. The Bullet Cluster (and for that matter the microwave background) behave just as they should if there is dark matter, and not at all as you would expect if gravity were modified. Any theory of modified gravity must have the feature that essentially all of its predictions are exactly what dark matter would predict. So if you want to convince anyone to read your long and complicated paper arguing in favor of modified gravity, you have a barrier to overcome. These folks aren’t crackpots, but they still face the challenge laid out in the alternative science respectability checklist: “Understand, and make a good-faith effort to confront, the fundamental objections to your claims within established science.” Tell me right up front exactly how your theory explains how a force can point somewhere other than in the direction of its source, and why your theory miraculously reproduces all of the predictions of the dark matter idea (which is, at heart, extraordinarily simple: there is some collisionless non-relativistic particle with a certain density).
And people just don’t do that. They want to believe in modified gravity, and are willing to jump through all sorts of hoops and bend into uncomfortable contortions to make it work. You might say that more mainstream people want to believe in dark matter, and are therefore just as prejudiced. But you’d be laboring under the handicap of being incorrect. Any of us would love to discover a modification of Einstein’s equations, and we talk about it all the time. As a personal preference, I think it would be immeasurably more interesting if cosmological dynamics could be explained by modifying gravity rather than inventing some dumb old particle.
But the data say otherwise. So most of us suck it up and get on with our lives. Don’t get me wrong: I’m happy that some people are continuing to work on a long-shot possibility such as replacing dark matter with modified gravity. But it’s really a long shot at this point. There is a tremendous presumption against it, and you would have to have a correspondingly tremendous theory to get people interested in the possibility. I don’t think it’s worth writing news stories about, in particular: it gives people who don’t have the background to know any better the idea that more or less everything is still up for grabs. But we do learn things and make progress, and at this point it’s completely respectable to say that we’ve learned that dark matter exists. Not what all of us were rooting for, but the universe is notoriously uninterested in adapting its behavior to conform to our wishes.
Brian, I think you’re making a good argument for why insisting on thinking of things this way can lead to the wrong conclusion. To answer your specific question, the mass as measured at infinity (the ADM mass, to be specific) does not change — it’s option (1), not option (2). Nevertheless, the pressure did increase. The resolution of these two statements is that it’s not quite right to think of “density plus three times the pressure” as the source of gravity. The stress-energy tensor is the source of gravity, and for the problem you have in mind you would have to derive the interior solution to the neutron star in general relativity and then match it onto the exterior Schwarzschild metric. Birkhoff’s theorem does indeed guarantee that the exterior metric won’t change during the process.
John Merryman,
To be frank, I don’t think much of what you said has any application to reality. Energy is the conserved current that results when certain properties of the system in question are invariant with respect to time. Energy no more goes back in time than you or I do.
The arrow of time, by the way, is indicated by thermodynamics. That is, entropy increases with time. This is the case whether you’re talking about entropy stored in radiation, or entropy stored in ordinary matter. The emission of radiation by a gravitationally-collapsed object is in no way, shape, or form the time reversal of gravitational collapse. Normal matter and photons just interact differently with gravity, that is all.
Paul, Brian:
What mass-energy density plus three times pressure gives you is the instantaneous rate of deceleration (i.e. the opposite of the second time derivative) of volume, for an infinitesimal unit volume of test particles that are initially at rest wrt each other. For homogeneous, isotropic solutions this fact can get you quite far, but when you have a localised object surrounded by vacuum, it takes a lot more work to see what that implies. (And that “three times pressure” is replaced by “the sum of the three orthogonal pressures” in more general situations.)
Have a look at Baez and Bunn’s
“The Meaning of Einstein’s Equations”
Jason,
It’s not a matter of not understanding what I said, you simply didn’t bother to read it through. I said energy goes forward in time, as in going from previous events to succeeding ones, ie. cause and effect. It is the events which go from being in the future to being in the past, as in tomorrow will be yesterday in two days. The arrow of time goes from what comes first to what comes second. For the event it is future to past. To the hands of the clock, the face is going counterclockwise. Prior to the Big Bang(in the context of BBT) the universe was in the future. Eventually it is presumed that it will be in the past. Future to past! It’s not really hard to figure out, if you think about it for a few seconds.
Entropy refers to usable energy in a closed system. A closed system is a unit in decline. It is losing more energy then it is aquiring.
Okay, I think I understand a little bit better now, but I still don’t see how it has any meaning. After all, events don’t travel from future to past: the perceived now is what travels forward in time.
Does anyone have answers or thoughts about the issue of gravity around interpenetrating pencils of light? tx
So the way to rule out the modified-gravity hypothesis is to find a system in which the dark matter and ordinary matter are located in separate places. If you see a gravitational force pointing at something other than the ordinary matter, dark matter remains the only reasonable explanation. And that’s precisely what the Bullet Cluster gives you. Dark matter that has been dynamically separated from the ordinary matter, and indeed you measure the gravitational force (using weak lensing) and find that it points toward the dark matter, not toward the ordinary matter.
Hi, a bit of a random question but:
How likely is it that gravity-wave detectors, say a future version of something like LISA, might eventually allow us to specifically observe the locations and quantities of dark matter involved in formations like this? Although, as you note, once dark matter has been compellingly demonstrated to exist in one case (as in has in the bullet cluster) there seems to be very little reason to suspect alternate mechanisms in other cases, it would still be nice to be able to confirm dark matter’s presence by something other than its interactions– so that, for example, we don’t wind up fooled by some situation like the Vulcan/Neptune split you mention, where some astronomical features explainable by dark matter do (like the Bullet Cluster) turn out to be caused by dark matter, but others turn out to be caused by by some other mechanism.
Might gravity-wave detectors eventually give us a way of directly observing dark matter in this way? Or is this unrealistic, or for some reason inherently not as powerful as existing techniques for detecting the presence of mass (like lensing)?
Sean–
Thank you for your quick response. That has been bugging me all weekend. It wasn’t entirely obvious to me that M would be conserved, since energy conservation has its own subtleties in GR, as I understand it. But now that I know that it holds for spherically symmetric collapse, you are right that it is a good example that the source of gravity isn’t simply rho + 3p.
Greg Egan–
I suppose I was getting confused by misapplying that local condition to the gravity of the entire star. I’ve printed out “The Meaning of Einstein’s Equations” to read later.
Paul Stankus–
In case you are still wondering, Cosmological Physics was published in 1999, and rho referred to mass density. Looking back, though, I don’t think the (1 + u^2/c^2) factor arises because the particles have extra gravity, in the sense of inducing more curvature in space-time, but because that’s the way the geodesics work out in the Schwarzschild metric.
Greg Egan — (replying to #102 above)
Thanks very much! for the description in terms of volume change and the pointer to Baez and Bunn. This “plain English” explanation of the meaning of the Einstein equations is just the sort of thing I’d been hoping to find, and at first glance seems to involve the roles of energy density and pressure fairly symmetrically which is very encouraging. However, afer having read through B&B’s pages I have to say — even though I am but a Bear of Little Brain — that B&B’s formulation appears to arrive at the wrong answer for the example we’ve been discussing.
You wrote that “…when you have a localised object surrounded by vacuum, it takes a lot more work…,” but B&B appear to handle this case quite directly in their page here
http://math.ucr.edu/home/baez/einstein/node6a.html
which explains how to recover Newton’s law in the weak field limit. Their derivation basically says that the second time derivative of the total volume occupied by a surrounding sphere of free-falling test particles is proportional to the integral of mass/energy density over the volume of the object, ie its total mass; and from simple geometry the a~1/r^2 law falls out immediately. I thought it was extremely cool and elegant!
The difficulty arises when we consider the case when the object’s internal pressure is non-negligible. If we follow B&B’s basic formulation then the second time derivative of any small volume becomes V”/V~(rho+Px+Py+Pz), ie proportional to (rho+3P) if the pressure is isotropic at that point. Using exactly the same argument as above, we would then expect that the gravitational acceleration experienced by distant objects would be proportional to the integral of (rho+3P) over the volume of the object. Would you not agree?
We know, however, that this cannot be the right answer! As discussed above (see #38, 94, 95, 97, 100) the example of the exploding (or imploding) spherically symmetric object is definitive: changes in its internal pressure do _not_ result in changes in the pull on distant objects; not even a little bit, but zip, zero, nada. So nothing like the volume integral over pressure can come into the answer for the pull on distant objects, as B&B seem to be stating. And, considering that the pressure profile within an object can be dialed up or down almost arbitrarily (within limits) _without_ causing any change in its apparent gravity, it is hard to see how pressure can make _any_ appearance in the calculation at all.
So, while B&B’s approach is certainly fun and does address a number of cases very elegantly, it seems to be just plain wrong on this question of how the pressure within an object affects its gravity. Do you see an alternative?
Paul
PS I will try to address your earlier comment in #89, but I will have to read it over more carefully first.
Coin,
I don’t think so. Dark matter tends to be quite diffuse, while detectable gravitational wave emission requires pretty high accelerations, such as the merger of supermassive black holes in the cores of galaxies. Our best bet is likely to remain gravitational lensing of background objects.
Jason: I see, thanks.
Paul (#108)
Baez & Bunn’s formulation of the Einstein equation is never wrong, because it’s provably identical to the tensor equation itself. But it has to be applied very carefully, because the simplicity that comes from the fact that it’s expressed locally and infinitesimally makes it easy to forget that you’re working with curved spacetime, with all the subtleties that entails. When they derive Newton’s law, they make approximations which are justified for that scenario, but which would not be reasonable in any situation where the pressure was relativistically significant.
I should stress that I’m not advocating B&B as a magic tool that unlocks any problem in GR, however difficult. The basic insight is priceless for anyone who’s stared uncomprehendingly at Einstein’s equation, and the examples they give — the cosmological models, and the Newtonian approximation — are very elegant, but for anything much more complicated you really need the full apparatus of differential geometry. It’s not that B&B would ever give you a different answer, if applied with sufficient rigour — but to achieve that rigour in general you really need what amounts to the whole framework of connections and parallel transport.
The anomalies have been resolved. The primary fix is to use Voyager’s measurement of the masses of Uranus and Neptune. Another source of error is a suspicious bias in a single 1895-1905 observation catalogue.
Standish, E. M. 1993, Planet X: No Dynamical Evidence in the Optical Observations, Astronomical Journal, vol. 105, p. 2000-2006
Jason,
The question is as to what the nature of time is. Is it a fundamental dimension along which physical reality travels, similar to space? Or is it a consequence of energy fields moving about and interacting, similar to temperature?
If it is a fundamental dimension of events and we are really at no particular point on them, just as our position in space is subjective, that means the earth revolves, as it rotates around the sun because there is a series of events called days and years. On the other hand, if time is a consequence of motion, then days and years are produced by the revolutions of the earth, as it rotates around the sun.
If you are willing to consider that it may be the second description, then time as any form of percieved dimension actually goes future to past, because it is not so much the actual energy that we mentally process, but the information produced by it. As the old saying goes, we live forward and think backward.
Consider the example of Schrodinger’s cat; The quantum fluctuation, the mechanism to open the vial of poison, the cat, the door of the box, our eyeballs and then our brain, isn’t the energy going forward in time, but the information of future potential collapsing into past circumstance. So multiple realities are not being produced by quantum indeterminacy, because the timeline we are actually perceiving is going from future to past.
The brain of an insect is basically a thermometer, because it just measures immediate activity. Our brains sequence that activity. This is time.
How does one decide that Abell 520 is ‘a mess’ but the Bullet is somewhere where it is ‘clear what is going on’?
If you already believe in collisionless dark matter, which says that galaxies, not gas, track mass, then the Bullet fits nicely into a clear just-so explanation (up to questions about the unusual size of some velocities) – whereas the data on gravitational lensing in Abell 520 don’t. If you believe in interacting dark matter, it may easily be the other way round. What is ‘messy’ or ‘clear’ depends strongly on your expectations.
You need to compare the data on a level playing field. Now perhaps it is the case that one set of data has more significance than the other (better-measured shear, better treatment of systematics, etc.etc.), or that one cluster is objectively more complex and subject to uncertainties in reconstruction than the other. Then that gives you a reason to make one of them your ‘poster child’, and punt on the other.
‘I dunno, looks like a mess’ is not an answer if you’re defending a theory that should apply to all clusters. Where’s Julianne on this anyway?
BTW I find large scale structure much the most convincing CDM evidence at present: it was actually predicted. Cluster studies are still relatively young because of the significant uncertainties in weak lensing, and one needs a large number of clusters to get any real handle.
A520 is a mess because it has 5-6 separate galaxy concentrations within it (as opposed to 2 for the bullet cluster), and the plasma has so many bumps and wiggles in it that it’s hard to find anything you’d call a core. It’s likely the result of many (5-6) smaller clusters that have merged over the past couple billion years.
That bump in the middle that has everyone excited is less than 2 sigma over what you would expect for the plasma (which contains 10-20% of the mass of any given cluster). We really have to wait for better data (meaning HST mosaics with either a refurbished ACS or WFPC3) to detect with reasonable significance if an excess of matter really exists at that position.
I should add that if you read the A520 paper, and believe the lensing map is correct you have to explain not only how the bump in the middle comes to exist, but also why one of the larger galaxy concentrations (the one labeled 5 in the figure in that paper) has apparently no mass associated with it while the smaller galaxy concentrations still have lots of mass.
The easiest explanation – lots of noise in the lensing reconstruction.
The authors of the A520 paper and I actually have approved HST time to get the data that Doug suggested. It was scheduled for WFPC2, but the launch date for the refurbishment mission got pushed earlier, closing our observation window. They’re pushing the program to WFC3, which will be a step up from WFPC2. So, we’ll know in about a year.
John Merryman,
Ah, well, that’s easy then. Time is just another dimension, like space. But this does not mean that the time line is something real. Rather a time line is just the time coordinate that a particular observer sees: other observers may see other time coordinates. Basically, since the advent of relativity, it’s been apparent that time and space were but different aspects of the same thing, space-time.
Granted, the properties of space-time may be an emergent property of some more fundamental, microscopic physics. Presumably discovering quantum gravity will tell us this. But, as far as all of the physics we know to date is concerned, everything plays out its part against a background of space-time, and that space-time is not an emergent property of any of the physics we know (e.g. photons, protons, neutrons, electrons).
Jason,
You seem to have put the usual amount of thought into that.
So you’re willing to say there is a dimension of events and physical reality travels along it, rather then the motion of physical reality causes a series of events, as each is replaced by the next?
The reason time is relative to the effects of velocity and gravity is because the rate of atomic activity is affected, so that the rate of change is variable. Atomic(and molecular) activity is temperature. As you pointed out, one of the most elemental arrows of time is thermodynamics. So it would seem temperature is more elemental than time.
Is temperature caused by motion, or is there a temperature scale which provides the framework for physical activity?
Are you really sure that describing the process of time in terms of a line isn’t intuitively based on our own linear motion.
You still seem to be thinking that time is some sort of absolute thing, marching from past to future in a deterministic manner. This isn’t the case, as how that passage of time is perceived is different depending upon the motion of the observer.
That is, a change in the appearance of the passage of time can be quite dramatic merely from changing our coordinates: the underlying physics is the same, the relationship between various particles is the same, but it [i]looks[/i] entirely different. Changing the temperature of a system, on the other hand, is a physical change to the system. It’s not just a different observer seeing the same thing done differently, but fundamentally different physical interactions are possible at different temperatures.
But fundamentally, the problem I have with your description is that it’s not mathematical in nature. The reason why I say that time is a dimension on the same footing as the spatial dimensions is that General Relativity has been shown to be a highly accurate theory, and within the mathematics of GR, we find that time [i]is[/i] another dimension, which shows us, beyond a reasonable doubt, that even if there is an alternate description where time is emergent, then it also must be the case that space is emergent from the exact same physics.
One slight correction: time does, of course, march from past to future in a deterministic manner. Perhaps a better word for what I was trying to say is that it does not march from past to future in an absolute manner.
And sorry for using UBB instead of HTML code. Old habits die hard.
Jason,
What have I said that makes you think I’m describing time as absolute, or even deterministic? If I’m putting it a catagory similar to temperature, does that mean I’m describing temperature as absolute? The only absolute in temperature is the absence of any motion and the same applies to time.
Describing time as a dimension is a natural assumption. History is essentially viewed as a timeline. Just as it is mathematically convenient to describe temperature as a scale, doesn’t mean there is a existent Platonic form of a temperature scale.
Process may be deterministic and time is process, not dimension.
Temperature is an absolute scale, though. That is, a substance at, say, 100K will have entirely different physical behavior than that same substance at 10,000K. So changing the appearance of passage of time by transforming your coordinates to another observer is in no way like temperature, because what you see is merely a different mathematical description of the exact same physical phenomenon. With different temperatures, you not only have a different mathematical description, but also a different physical phenomenon.
Now, there are some parallels, such as with the path integral formulation of quantum mechanics versus the partition function in statistical mechanics, where we find that a path integral in quantum mechanics is all the exact same math as the partition function with the following replacement:
kT = h-bar/(i*t)
..where i is the square root of -1, t is the time, k is Boltzman’s constant, T is the temperature, and h-bar is Planck’s constant divided by 2pi.
Some have proposed that there might be some fundamental reason for this connection, but none has yet been found (so far as I know), and if there is a connection, it’s not going to be simple, intuitive, or obvious. However, it would probably be elegant in a mathematical way.
Jason,
This is not an analogous comparision. Since you are comparing the same phenomena in time from different perspectives, a proper analogy would be that from the perspective of someone at 100F, 75F would seem cool, while to someone at 50F, 75F would seem warm.
I didn’t say temperature and time are the same thing. I said they are both methods of describing motion. One is the level of activity against a prevailing scale. The other is rate of change relative to context. Because there is implied direction of change doesn’t mean it has absolute dimension, because in the absolute all effects cancel out. Entropy is a process of cancelling the substance of the closed set, so that it reaches thermal equilibrium with the larger context. “For every action there is an equal and opposite reaction.” To the hands of the clock, the face(context) goes counterclockwise.
You haven’t directly addressed the point that if time is the basis of motion, then physical reality travels along it, from past events to future ones, so that it is the existence of days(dimension of time) which cause the rotation of the earth, rather then time being a function of motion, so that the rotation of the earth creates days. So here is the question; Which is cause and which is effect? Rotation(motion), vs. days(time).