There has been a fair amount of buzz recently about Erik Verlinde’s latest paper on the arXiv called “On the Origin of Gravity and the Laws of Newton” from his talk on December 8th,  2009 at the Dutch Spinoza-instituut.  There is no denying that this guy knows his stuff.  That being said, there is a lot of chatter about the validity of some of his arguments, and I am inclined to agree with some of the naysayers.  EDIT: Since writing this, Erik Verlinde has posted comments to address some of these points raised with the Logic of the paper.

This paper demonstrates a kind of logic very commonly found in the more philosophically oriented extensions of general relativity.

1. Start with Newtonian mechanics and obtain desired result
2. Extend results using the above logic to the relativistic domain

The logic seems obvious, from a historical point of view.  First, we had Newton, and then we had Einstein.  Shouldn’t future works follow that same evolution: a theory must begin with Newtonian mechanics before seeing results akin to General Relativity?

I would argue “no”.  Logic that applies in one domain by no means applies in another.  From the trivial perspective, Newtonian mechanics just seems to slide nicely into Special and General Relativity, but mathematically, that isn’t the case at all.  Newtonian Mechanics applies on, nice, simple Euclidean manifolds of any dimension.  General Relativity applies on Lorentzian manifolds, of dimension greater than two.  For the reader who doesn’t think the manifold matters that much (other than maybe having a different metric), I won’t have time to go into the depth here required to convince you, but I encourage you to think on it (or tell me why you think it doesn’t matter).

We’ll start by making our way through Verlinde’s Introduction. I’m going to skip some of my usual minor objections, like things “carrying energy”, and try to focus on, what appears to be the logical crux of this work that I am disagreeing with.  Note: I’m not going to attacking holography or anything of the sort, this is simply a language/logic objection.

From the first paragraph:

The universal nature of gravity is also demonstrated by the fact that its basic equations closely resemble the laws of thermodynamics and hydrodynamics.

I would probably be one of the last people to disagree with the “universal nature of gravity”, but claiming that it is because the equations, in one particular form, resemble the laws of thermodynamics and hydrodynamics is quite unreasonable.  There are several reasons that those particular forms of the equations look the way they do, the main one is, because we wanted them to look familiar.  General relativity and thermo/hydrodynamics share similar mathematical techniques, and why shouldn’t they? We created those techniques to be useful.  If things work well in one area, we would be crazy not to try to apply them to other areas.  Verlinde claims that there has not been an explanation for these similarities, but I disagree.  Equations can take many forms; the fact that they both use tensors is no spooky coincidence.  The mathematics may look similar, symbolically, but it is representing something incredibly different.  We describe gravity in a four-dimensional spacetime, but we do not describe heat or entropy in one (okay, sometimes we do when talking about black holes and the like, but that’s actually a totally different matter).  Even if symbolically they look similar, the mathematical background those symbols exist in is profoundly different.  We, as humans, created the mathematics (mathematical Platonists, feel free to jump on this) and created the symbols, and we like to arrange them in ways we know how to work with.  That is all this resemblance is.

Verlinde is coming from string theory, so his push for a strong holographic principle should be no surprise.  Within string theory, this principle has been very useful in solving a few paradoxes, ie. with black holes.

By reversing the logic that lead people from the laws of gravity to holography, we will obtain a much sharper and even simpler picture of what gravity is. For instance, it clarifies why gravity allows an action at a distance even when there in no mediating force field.

Now remember black holes come from General Relativity, where there already was no problem with “action at a distance” or a need for “force fields” (because remember, gravity is not a force in GR).  The holographic principle would not exist if it wasn’t for General Relativity and its stringy extensions.  That being said, it’s a great idea to try and see if the holographic principle yields an extended General Relativity back.  But, being surprised that it gives relativistic results, doesn’t make sense.  Admittedly, stringy complications do muck up the beauty that is standard GR, but those particular “clarifications” weren’t needed.

A problem that is practically impossible to avoid when doing this kind of thing, is getting too caught up in believing in “the truth” of your assumptions.  Verlinde is starting with the assumption that space (without gravity) is emergent.  There is plenty of room to disagree with this here, but let’s push on for now.

From Section 3.  Emergence of Newton’s Laws:

Space is in the first place a device introduced to describe the positions and movements of particles. Space is therefore literally just a storage space for information.

If the first sentence is true, then it’s hard to argue with the second sentence.  That being said, the first sentence is only true in the Newtonian picture.  Space, to a relativist, is much much more.  Since Verlinde wants to recover Classical Mechanics, he defines information to be stored on “screens” (separations between points in space). Whether this really seems natural, I can’t say.  Now comes a problem that I don’t honestly know how to properly address.  Verlinde uses an analogy between Bekenstein’s thought experiment that lead to black hole thermodynamics and his holographic screens (“We want to mimic this reasoning not near a black hole horizon, but in flat non-relativistic space”).  You can read the details in his paper and make your own mind up as to the validity of the analogy.

Now we see thermodynamics described for the screen (from analogy with a black hole event horizon… which was defined from analogy with standard thermodynamics…), so we are ready for “force” to emerge:

How does force arise? The basic idea is to use the analogy with osmosis across a semi-permeable membrane. When a particle has an entropic reason to be on one side of the membrane and the membrane carries a temperature, it will experience an effective force equal to

This is the entropic force.

So we had defined change of entropy for our screen, and that lead us to a Newtonian force.  Now, to me, it seems that it only was able to lead us to such a force because we already knew the Newtonian definitions.  Force didn’t arise, we just continued the analogy.  The fact that we got evidence of Newtonian Mechanics isn’t remotely surprising here nor does it seem significant.   Verlinde continues this logic through to arrive at the Newtonian definition of gravitational force.

We have recovered Newton’s law of gravitation, practically from first principles!

Particles here were already assumed to have classical mass and energy associated with them, and thermodynamics was fully defined, so the fact that Newton’s laws were recovered really should not be a surprise, as they were basically implicit in this exercise to begin with.

Interesting, Verlinde goes on to acknowledge the fact that Newton’s laws were already party of the process, but then seems to draw a different conclusion:

These equations do not just come out by accident. It had to work, partly for dimensional reasons, and also because the laws of Newton have been ingredients in the steps that lead to black hole thermodynamics and the holographic principle. In a sense we have reversed these arguments. But the logic is clearly different, and sheds new light on the origin of gravity: it is an entropic force! That is the main statement, which is new and has not been made before. If true, this should have profound consequences.

I’ve read this paper a few times now, and I still can’t see his point.

The logic of the argument, as I see it

In the next section, Verlinde summarizes his assumptions:

Our starting point was that space has one emergent holographic direction. The additional ingredients were that

(i) there is a change of entropy in the emergent direction

(ii) the number of degrees of freedom are proportional to the area of the screen, and

(iii) the energy is evenly distributed over these degrees of freedom.

I still think he left off (iv), which was “Newtonian Mechanics”.

Why do we need the speed of light c in this non relativistic context? It was necessary to translate the mass M in to an energy, which provides the heat bath required for the entropic force.

Which we really only know from the relativistic context.

[Space] has to be endowed by a book keeping device that keeps track of the amount of information for a given energy distribution. It turns out, that in a non relativistic situation this device is provided by Newton’s potential. And the resulting entropic force is called gravity.

It needs to be pointed out also that Verlinde only derived equations with the appearance of Newton’s laws.  For a truly emergent space, we have to define absolutely everything: What does position mean? What is velocity? What is energy? The equations are only meaningful, because we equate them with the well defined equations of motion from Newton.  Equations, without definition of background, don’t actually define anything.  They are just symbols, that we can only give meaning to, by our initial assumption of their meaning.  In section on the Einstein equations, Verlinde assumes again an emergent spacetime and then uses the relativistic analogues for the equations in the Newtonian section to arrive at the Einstein Equations in the same manner.

I’ve included some resources as the bottom, and I encourage anyone who is interested to read the actual paper themselves, because it has a lot of content in it and some very clever ideas.  That being said, to me, it reads as something that starts with the desired conclusions so firmly in mind, that it is bound to end up with them.  If you want space to be emergent, then that is great, but then don’t let it take all of the properties of Newton’s space before you even begin.

Resources:

Website of Erik Verlinde

On the Origin of Gravity and the Laws of Newton – arXiv:1001.0785v1

Black holes and Entropy – Phys. Rev. D 7, 2333–2346

Thermodynamics of Spacetime: The Einstein equation of state – Phys. Rev. Lett. 75, 1260–1263

Johannes Koelman’s Remarks: It From Bit: The Case Of Gravity

Wikipedia: The Holographic Principle

Wikipedia: Black hole thermodynamics

Edit: Some might say that Newtonian Mechanics was not implicit in the derivation because “thermodynamics does not need Newtonian mechanics”.  That I both agree and disagree with.  Many aspects of thermodynamics can exist wonderfully in isolation from Newtonian Mechanics, using just definitions of microstates and a mechanics free definition of energy.  However, it still appears that a Newtonian background was implicit in part of the discussed derivation: without the Newtonian concept of “force” in mind, the conclusion , wouldn’t have been meaningful.  If this is supposed to be an emergent space, than we don’t get the luxury of well defined concepts like “acceleration”, without a price.  Assumptions have to be made, and it appears that the assumptions that lead from thermodynamics to the discovery of the second law of Newton where already those from Newtonian mechanics.

Edit: Verlinde’s response: Logic of the paper.

Good blog discussions on Woit’s Not Even Wrong and Motl’s The Reference Frame.

Further Update: There is some superb analysis and comments over at Motl’s blog

NASA guy Roger Weiss emailed me the link to this video, on Space.com, and I thought I might as well make my comments here.

How to Time Travel?

Despite being really solid physicists (to whom I could probably listen to talk for hours), Charles Liu and Michio Kaku make a few of those misleading, “popularizer”, statements that I just want to point out.  The problem with much of the popular work done in physics is that you only can get the real picture, if you already know what they are talking about.  The lay audience for whom popular work is intended, thus misses out on the key concepts.  Very important questions like, “whose time are we even talking about”, are not even mentioned in these videos.

Michio Kaku:  …Space and time is a fabric – a fabric that, perhaps, you can rip, if you have unimaginable sources of energy.

I object to this “ripping” of spacetime.  Time travel, whether you believe the mathematics that supports it are physical or not, is something that exists within spacetime (as does everything else).  Travelling backwards in time, just like travelling forwards in time (which I’m doing right now), is just moving to a different part of the spacetime manifold (albeit the causal structure of that manifold, classically, forbids much of the possible movements).  The analogy with ripping a piece of fabric is very misleading.  Even a wormhole is not a “rip” in spacetime, it is just a topological feature in spacetime.  Ripping a piece of fabric, implies that you are holding a piece of fabric and ripping it from the outside.  This is completely acausal, when considering the universe.  You can not take and rip spacetime, because to do so, within the fabric analogy, would imply the existence of action outside of spacetime.   We can not rip or “split open” spacetime – spacetime is just spacetime – nothing exists outside of it or acts on it as a whole (ignoring possible inflationary themed pocket universes (other spacetimes) colliding with our own).  It can have an absolutely bizarre shape;  it can be curved and warped and have holes (topologically), but can not be “ripped”, because “ripping” implies an “outside” of spacetime.

Charles Liu:  …The idea of a wormhole is, very simply, that, you come into that hole from within our universe, temporally exit the universe, exit our spacetime continuum, and return again to our spacetime continuum at some other space location and some other time location.

The idea that a wormhole leads you out of our spacetime, while easier to visualize, is incorrect.  Within general relativity, you never leave spacetime, regardless of what you do.  The misconception comes from the problem that people seem to have with embedding.  Because it’s so difficult (read: impossible) to visualize a curved 4-dimensional manifold representing spacetime, we choose to imagine a 4-dimensional manifold embedded into a 5-dimensional Euclidean space.  For an example, see the analogy:

Analogy to a wormhole in a curved 2D space - this image is used in Part 2 of the video

Here, this 2-dimensional image (because it’s a picture) is representing a 2-dimensional spacetime embedded into a flat 3-dimensional space.  Travelling through the wormhole looks as if you are leaving the 2-dimensional spacetime, and entering the 3-dimensional flat space.  This is not what is actually happening.  The 3-dimensional space is just a tool for visualization, there is nothing real about it.  The entire universe here is the 2-dimensional spacetime, which includes the wormhole.  Embedding diagrams are nice aids, because most of us would rather imagine a curved surface within a larger dimensional flat manifold than just a curved manifold, but they mislead us.  Read more about embedding diagrams here.  (There are also issues with using 2-dimensional images to help us think about 4-dimensional spaces, but that can be for later). In brief: you are not exiting the universe/spacetime continuum at all when going through a wormhole – it isn’t meaningful to talk about “exiting” something that is the entirety of everything.  There is nothing to exit to – our universe (within general relativity) is not actually embedded into anything, it’s just curved.

Michio Kaku: Travelling to the future is easy, our astronauts do it all the time.

Me too? “Interestingly”, one doesn’t need a spaceship to travel to their own future,  my chair suffices for me.  I honestly not completely sure what he was going for with this… But on to the second, shorter, video.

Can You Time-Travel? (How to Time Travel? Part 2)

The joys, terrors and true possibilities of navigating the Fourth Dimension, with quantum physicist Michio Kaku and astrophysicist Charles Liu.

Michio Kaku: …well, Plutonium does not have energy to drive a time machine.  To energize a time machine, to bend time into a pretzel, to punch a hole in the fabric of space and time, would require the energy of a star.  One version of a time machine uses what is called, a wormhole.  Think of the looking-glass in Alice and Wonderland.  That looking-glass is the wormhole.

Honestly, I’m not sure what to make of this discussion.  Travelling backwards within one’s own time just requires spacetime to have the time-dimension be a closed loop, not necessarily a pretzel (which has a much more complicated topology than required) – although I assume he picked “pretzel” for poetic license, because it sounds nicer than “ring” or “circle” or  something else with the same fundamental group as .  My objection to punching a hole in spacetime is the same as my earlier objections.

As for the energy required to use a worm hole to travel backwards in time… I honestly am not sure where Kaku is coming up with that, so I can’t comment to the specifics.  As far as I know, there are no confirmed estimates to how much energy one would need to use a wormhole for travel back in one’s own time.

The wormhole-looking-glass analogy, I do have a problem with.  In Lewis Carroll’s Through the Looking-Glass, Alice passes through a looking-glass, showing her reflection, into an alternate world.  That really has nothing to do with what a wormhole is.  A wormhole, in general relativity, is just a “short cut” through spacetime.  It doesn’t take you to an alternate universe, the laws of physics aren’t changed – it’s just a non-trivial connection between two points in spacetime.  The looking-glass that Alice went through didn’t take her to another point in space and time, it took her to an alternate universe.

The way we talk about spacetime, and the tools that we use to help us visualize it, can do a disservice to our understanding of the universe.  It’s nice to be able to talk about the universe as a whole, as if we were looking at it from the outside, but to visualize the whole universe – to take into account any theory that requires knowledge of the entirety of spacetime – is to throw out causality (assuming the universe is even reasonably large).  This disregard for causality is much more substantial than that allowed for by time travel.   To be able to perceive all of spacetime, is to know all future and past events in the entire universe – to be omniscient.  So in order to be omniscient, one must be outside of spacetime.  But in general relativity, there is nothing outside of spacetime.  If we allow for physics to be discussed in a way where omniscience is possible (by imaging that this embedding is something real), we aren’t doing physics (as it currently stands) anymore.  If we want wormholes to lead us from our universe into another universe, to preserve causality and timelines, then we really aren’t using the wormholes of general relativity, but imagining some other object.  That’s fine, but we should call them as such, and not give properties to a mathematical consequence of general relativity that simply aren’t there.

-S.C. Kavassalis

EDIT: As a minor note, it seems I watched the videos out of order.  It doesn’t matter much for continuity though, as they both are fine as stand-alones.

Physicists: Stop saying “gravitational force” or that a body is”accelerated under/by gravity.”

This objection is only true within the frame of general relativity – when using classical mechanics (to mean Newtonian or non-relativistic quantum mechanics), you can say “gravitational force” or “Newtonian force” as much as you would like to refer to gravity, but within general relativity, it is not a meaningful term.

To really appeal to the English language (instead of just mathematical definitions for this one), the word “force” in physics, according to 2009 Random House Dictionary means:

[A]n influence on a body or system, producing or tending to produce a change in movement or in shape or other effects.

More specifically, the 2009 American Heritage Dictionary gives the definition of a force in physics as:

A vector quantity that tends to produce an acceleration of a body in the direction of its application.

For those familiar with the description of gravity due to general relativity, it should be clear then, that gravity is not a force, in either of the above two senses.  The motivation behind general relativity is to no longer describe gravity as a force: gravity is a consequence of the geometry (specifically meaning the curvature) of spacetime. Particles travel along geodesics (locally straight paths) through the curved spacetime. As far as they are concerned, they aren’t being accelerated (ie. being acted on by a force); they are just travelling along inertial paths (the same as a book sliding across a frictionless table).

For a satellite falling into a planet, one shouldn’t say that it is “being accelerated around the planet by the force of gravity due to the planet” but we should say that it is “travelling along a locally straight path that, to an observer in a non-inertial frame, appears to be curved because of the curvature in spacetime due to the matter of the planet” (although one could probably word it a bit more nicely).

If the satellite turned on a rocket and left its geodesic, then it would be acted on by a force, but that force would be due to the rocket. In the same way, in the Newtonian definition, you wouldn’t say that cart moving with constant velocity (on a frictionless track) was being acted on by a force; you shouldn’t say that an inertial body moving along a geodesic through spacetime is being acted on by a force, whether it’s a satellite in orbit around a planet or a particle falling into a black hole (to them, they’re just going straight).  When you throw a ball and it falls in a parabolic arc, remember it’s spacetime that is curved, not the particle’s trajectory (unless it really is being acted on by another force). It just looks curved to us, because we are in a non-inertial frame (when we’re standing watching the ball, the Earth is exherting a force on us, so we aren’t following a geodesic relative to the Earth, unlike the ball).

Despite the relative simplicity of this concept, many people continue to refer to gravity as a force, within the scope of general relativity, when, by definition of English or of relativist, it is not. In quantum gravity, it may be the case to again treat gravity as a force, but in terms of general relativity, it is an inappropriate use of language, and really goes against the basic “equivalence” motivation behind general relativity.

One example in the literature:

“Hawking Radiation As Tunneling” by Maulik K. Parikh and Frank Wilczek (Phys. Rev. Lett. 85, 5042 (2000). Cited 215 times ).

When considered at the very broadest level, radiation of mass from a black hole resembles tunneling of electric charge off a charged conducting sphere…For while the electric force between like charges is repulsive, the gravitational force is always attractive.

Despite an otherwise very formal (and brilliant) paper, gravity is still paired, analogously, with the electromagnetic force. Within general relativity, this is simply not a valid analogy. Although it sometimes helps to give physical motivation to problems by considering them in terms of our classical pictures, it shouldn’t be the basis for anything (especially when the actual explicit work of the paper is done within proper general relativity). While tunneling may in fact be the mechanism for Hawking radiation, its motivation comes from a faulty analogy.

If it wasn’t for our familarity with Newtownian forces, removing that usage of force from the relativist vocabulary wouldn’t be so difficult. Why use an expression that is actually not meaningful within a field (and is in fact contradictory to the basic principles behind it)?

-S.C. Kavassalis

“Black Holes – a Simplified Theory for Quantum Gravity Non-Specialists” by Vladan Panković (available online: http://arxiv.org/PS_cache/arxiv/pdf/0911/0911.1026v1.pdf)

My first thought on reading the abstract for this paper is, “What is the point?”  I’m all for bringing theoretical physics to a more general audience, but never at the expense of the content of the theory.  While this paper is hard to read because of translation issues (and even as a language snob, I can’t actually fault people who are writing outside of their native language), more importantly, it does a disservice to physics, by incorrectly presenting ideas, in an attempt to simplify them.  The intention however of Pankovic’s paper is good, to present a simplified account of the Kerr-Newman black hole solution to Einstein’s equation and the relevant thermodynamical properties (which I object to on another level, but that’s not the issue right now) to interested non-specialists.  The implementation is what I take issue with.

Introduction, first paragraph:

[A]ccurate analysis of the dynamical and thermo-dynamical characteristics of the black hole needs knowledge of the subtle details of general theory of relativity and quantum field theory, i.e. quantum gravity (even if there is no complete theory of the quantum gravity to this day).

Technically, the Newman-Kerr metric, found in 1965, is a description for a rotating and electrically charged black hole, the dynamics of which don’t require quantum gravity at all to explain (J. Math. Phys. 6, 915 (1965)).  Originally, black hole thermodynamics was a fairly “classical” phenomenon in terms of its description, but these days, the thermodynamics, as far as I know, are considered in light of the AdS/CFT correspondence, which of course does require quantum field theory and certain aspects of certain theories of quantum gravity.

Introduction, second paragraph:

In this work we shall present a simplified method for description and calculation of the Kerr-Newman black hole main dynamical and thermo-dynamical characteristics. This method is physically based on the well-known principles of the classical physics (mechanics, thermodynamics and electro-dynamics).

I admit that the second sentence here scares me a little bit. Presenting simplified descriptions in terms of phenomenology is one thing, but presenting descriptions of something that you admit requires not only general relativity, but a theory of quantum gravity, in terms of classical mechanics should be a warning sign to most readers.

Within this second paragraph, Panković lists the tools that he will use in his treatment (along with classical mechanics): “non-relativistic quantum mechanics”, “statistical mechanics”, “the elementary form of the general relativistic equivalence principle”, and “a linear approximation of the quantum gravity theories”.

There is no denying that these are the tools that black hole thermodynamics started out using, but thankfully (or unfortunately, depending on how you look at it), we have come a long way since then (with the exception of the approximation of quantum gravity, which I haven’t a clue what could mean here).

Part 2 – Dynamics: Panković sets up an analogy with the Schwarzschild vacuum solution and Laplace’s dark stars (classical objects that obey Newtonian mechanics).  Both objects have similar properties at first glance, but there are some important and fundamental differences.

We’ll skip the full discussion of the Schwarzschild vacuum metric, and just begin with the expression of the line element (in Schwarzschild coordinates), after taking the weak-field approximation.

Now, remembering that Schwarzschild coordinates are just a choice of coordinate system, we can choose to redefine our , in order to agree with our Newtonian familiarities as,

This is the representation of the Schwarzschild radius that people are most often familiar with and the one that Panković is making the analogy to the “dark stars” with.  It’s important to remember the derivation of this line element though, and the assumptions that are made in the process.  Mass does not appear in the derivation until the very end, and is really only included for sake of analogy with Newtownian mechanics.  There is nothing fundamental about the meaning of mass in the above expression, other than that we are adding in physical, Newtonian constants, during the weak-field approximation part of the process.  In actuality, there is no mass in the Schwarzschild vacuum solution.

This radius corresponds to the critical radius that an ideal Schwarzschild black hole, that we are ascribing mass to, would have such that the escape velocity of in-falling particles would have to be greater than or equal to the speed of light.

Dark stars have the same critical radius, although the interpretation of the two should be quite different.  Starting from Newtonian mechanics, Panković shows the derivation of the critical mass for the dark star with mass , by starting with the assumption that the total energy of the local system (classical translation kinetic energy of the in-falling particle plus the negative potential energy of the Newtonian body) is zero.  One obtains the same expression for the radius, such that an in-falling particle, with velocity less than the speed of light, could not escape the gravitational pull.

In this way presented Laplace’s method (equation) can be considered as an extremely simplified method for determination of the Schwarzschild black hole horizon as the basic dynamical characteristic of the Schwarzschild black hole.

The fact that one obtains in both cases (in coordinate systems that happen to be somewhat locally equivalent), should not suggest that Laplace’s method should be used as an analogy, at all, for a Schwarzschild black hole.  A very important thing to remember: The dark star calculations are done for a three-dimensional Euclidean space, while the black hole calculations are done for a four-dimensional Lorentzian spacetime (there is a big difference between the two).

Dark stars have a finite radius, a Schwarzschild black hole does not (the centre is a singularity).  Newtonian mass is part of the definition of a dark star, and is only labelled as such in a Schwarzschild black hole for familiarity’s sake.  The derivation of a dark star’s critical radius is based on the total energy of the system of star and probe particle being zero, energy isn’t even remotely treated in this way in general relativity (energy for probe particles is reference frame dependent and the analogy of “gravitational potential energy” (ie. geometry) in general relativity is a combination of energy, mass, momentum, pressure, and tension, and if we were really considering a star like in the classical picture, we’d have to take binding energy into an account).  Dark stars can classically emit indirect radiation, a Schwarzschild black hole can not.  Dark stars also do not have the ability to bend light to the same degree as objects in general relativity, like Schwarzschild black holes do.

Ie. a dark star is not a good analogy for a Schwarzschild black hole.

Next, Panković goes on to modify the calculation for a static dark star to one that is rotating and electrically charged (building to his Newman-Kerr analogy).  Similar problems arise as with the Schwarzschild case.  The solution for a Kerr black hole literally describes a ring singularity, not a rotating, spherical, star (although it is the best classical representation for the area outside of a rotating massive object).  Classically, a point particle cannot have any angular momentum (because there can be no defined axis of rotation due to the symmetry), so the singularity in the Kerr (and Newman-Kerr) solution must take the shape of .  As far as I can tell, Panković tries to include the energy of rotation of his probe particle (whose rotation is induced by the rotating star), but not that of the dark star itself.

Namely, classical mechanical rigid body, with radius and homogeneously distributed over volume mass , holds momentum of the inertia but not. It implies classical angular momentum but not . In this way use of instead of in (4) represents an ad hoc postulated correction of the classical expression.

A few points: there are no real rigid bodies; angular momentum in the Kerr solution is measured from infinity (and how you measure angular momentum, just like energy, matters a great deal in general relativity); ad hoc changes to classical formula don’t make them equivalent to those from special or general relativity; if the star were exactly spherically symmetric, it would collapse into a Schwarzschild black hole, and then couldn’t be rotating (so using expressions for a perfect sphere aren’t necessarily reasonable).

The Kerr (and Newman-Kerr) solution gives us a rotating ring singularity surrounded by a spherical event horizon, somewhat similar to the event horizon found with the Schwarzschild vacuum solution, but the overall picture is much more complicated:

From Hans Stephani's "General relativity: an introduction to the theory of the gravitational field" (2nd edition)

There are many excellent references for people wanting to know the specifics for the Kerr solution, a couple good books are: “General relativity: an introduction to the theory of the gravitational field” by Stephani (introductory), and “The Large Scale Structure of Space-time” by Hawking and Ellis (more advanced).

Regardless of the vast number of fundamental differences between the Newman-Kerr solution and that of a charged, rotating dark star, Panković is able to use his method from the Schwarzschild case to derive an inner and outer event horizon (ie. and ), as in the proper Kerr solution.

Pankovic obtains:

,

in the same, spherical, coordinates as used in the “Schwarzschild” dark star case, where , for the probe particle.  Panković labels to be a distance at which the probe particle would rotate with peripheral speed (I hope the probe particle isn’t actually supposed to be a point particle in all of this).  In the Kerr solution, appears as a constant in Boyer-Linquist coordinates, where , where is the angular momentum of our rotating mass, .  These are not conceptually equivalent, nor are Boyer-Linquist coordinates equivalent to those used by Panković.  One should never even be talking about a massive particle having any component of it’s velocity equal to the speed of light, but this is what Panković is doing here.  In order to make the calculations easier to follow, he is neglecting special relativity (which should be very hard to justify, seeing as not only are his massive particles near the speed of light, but they are actually travelling at it).  The fact that equivalent equations turn up is due to carefully chosen coordinate systems.  The physics is still completely different.

Structurally though, for Boyer-Linquist , we do obtain the same values for an inner and outer radius but the coordinate systems are not the same.  How one decides that the values of are meaningful, in the dark star picture, I can not say.

Continuing the written discussion through the thermodynamics section does not feel overly useful at this point, when it seems that this paper isn’t on black holes at all, but on dark stars.  There is nothing wrong with a paper being on Newtonian physics, so long as that is what it claims to be.  Analogies between Newtonian physics and general relativity are often dangerous, because they lead to misconceptions (like just how significant “mass” really is  for the Schwarzschild solution).

Dark stars exist in space, not in spacetime; they are usually considered to be spherical with finite radius; they can radiate light out of their critical radius; they do not bend light around them in the same way that black holes do… trying the bridge the gap between Hawking radiation and a dark star seems entirely meaningless under these considerations.  Although both interesting, black holes (static or rotating and charged) are not analgous to dark stars.  Claiming they are analoguous isn’t doing either Newtownian mechanics or general relativity any justice.

In the final line of his conclusion, Panković claims:

…[A]pproximate method for the decription of the basic dynamical and thermodynamical characteristics of black hole can be very useful for the quantum gravity non-specialists.

I would have to disagree.  While interesting objects classically, dark stars seem to have no application to quantum gravity at this point (in the face of the full on neglect of the basic tenants of general relativity)

-S.C. Kavassalis

]]> http://badphysics.wordpress.com/2009/11/08/dark_stars/feed/ 2 S.C. Kavassalis Kerr Solution http://badphysics.wordpress.com/2009/10/21/metric/ http://badphysics.wordpress.com/2009/10/21/metric/#comments Wed, 21 Oct 2009 20:42:46 +0000 S.C. Kavassalis http://badphysics.wordpress.com/?p=21 ]]>

Physicists: Stop using the word “metric” to mean so many different things. A metric tensor is NOT the same object as a metric, it is NOT the same object as its matrix representation, and it is NOT the same object as its associated line element. You should not use those words interchangeably, they are not equivalent structures.

A metric is a function defined on a set.
A metric tensor is a tensor field.

If local coordinates are known:
The matrix representation of a metric tensor is a matrix.
The line element is a function of a metric.

In mathematics, the word metric refers to a fairly general function which defines ‘distance’ between elements in a set (it takes in elements of a set, and produces a real number). Riemannian and pseudo-Riemannian metrics (there are many more kinds of classification of metric too) have different conditions on those functions, but that’s more detail than is required here.

A metric tensor is a function defined on a manifold (a vector space) that takes in two tangent vectors and produces a scalar quantity. Metric tensors are used to define the angle between and length of tangent vectors (somewhat analogous to the dot product of vectors in Euclidean space)

Defining a metric versus a metric tensor

Consider a smooth manifold of dimension n. For every point x in our manifold, there is a vector space called a tangent space (a tangent space contains all of the tangent vectors to our manifold at the specific point x).

Now, a metric at our point x is a function g_x(X_x,Y_x), which takes in the two tangent vectors X_x and Y_x (at x), and outputs a real number. The metric function must also be bilinear, symmetric, and nondegenerate, but we don’t need to go into further details.

Now we can define a metric tensor, g, on our manifold: The metric tensor assigns a metric, g_x, to every point x in the manifold (such that it varies smoothly with x in the manifold). The metric tensor is then:

g(X,Y)(x) = g_x(X_x,Y_x)

For those familiar with tensors, it should be clear that the metric tensor is actually a tensor field (a tensor is assigned to each point of our mathematical space). A metric tensor is not the same as a metric (it’s more analogous to an ‘infinitesimal’ metric function), but it is usually understood in differential geometry and related areas in physics that when one says “metric”, they really mean “metric tensor”. Mathematically, they are not equivalent objects, but integration of a metric tensor does induce a metric function.

Most of the time when actually doing physics, we don’t want such a general object. If local coordinates are known, the metric tensor can be expressed in a variety of more useful forms.

If we are in a region of the manifold where we have defined a local coordinate system, ie. x^μ (where μ runs from 0 to 3), we can re-write our metric tensor [field] as:

g = g_μν dx^μ⊗dx^ν

where, g_μν are real-valued functions, and dx^μ are one-forms.

If we have local coordinates defined, we can then represent the metric tensor in matrix form, where, for our four-dimensional spacetime, we will have a 4×4 matrix with elements g_μν.

In our local coordinates, if we take dx^μ to be an infinitesimal coordinate displacement, we can write out a line element: ds² = g_μν dx^μdx^ν. The line element, we know, is incredibly useful, as it provides us with an invariant quantity and also imparts information about causal structure.

EDIT: A note from The Unapologetic Mathematician that I should add: “the metric tensor is a bilinear function of two vectors at a given point, while the line element is a quadratic function of a single vector. However, the polarization identities will allow you to recover the bilinear function from the quadratic one.”

Why does this matter? Well, for starters, general relativity is really all about your frame of reference and choice of coordinates. Some structures are unchanged regardless of your choice of coordinates (ie. the metric function & metric tensor), and some structures change with change in coordinates (ie. the matrix representation of a metric and the associated line element).

Just a couple of (well cited) offenders:

C. Brans and R. H. Dicke, Mach’s Principle and a Relativistic Theory of Gravitation . Phys. Rev. 124, 925 (1961), Cited 1,139 times.

As in general relativity the metric tensor is written as

g_ij = η_ij + h_ij …

EDIT: If I included more of the quote, it would have been obvious that local coordinates had already been chosen and they weren’t writing out a general metric tensor, but a coordinate specific object.  The reference is cited for context.  Yes, abstract index notation for tensors uses indices to indicate the type of tensor, rather than to indicate components in a particular basis.  Unfortunately, sometimes it is possible to forget if one is actually referencing components in a specific basis or the abstract tensor itself with this notation.

As I said above, g_ij is not the metric tensor, or a tensor at all, but a set of real-valued function specified for a local coordinate system (g_ij are also the matrix elements in the matrix representation – in those coordinates – of the metric tensor). The same goes for η_ij and h_ij as well.

___________________________

Tullio Regge and John A. Wheeler, Stability of a Schwarzschild Singularity . Phys. Rev. 108, 1063 (1957), Cited 476 times (two authors I respect immensely)

Schwarzchild found long ago the solution of Einstein equations for the metric around a fixed spherically symmetrical center-of-mass:

ds² = -(1-3m*/r)dT² + (1 – 2m*/r)^-1 dr² + r²(dθ+sin²θdφ²) …

This is the line element, not the metric.

___________________________

Brandon Carter, Global Structure of the Kerr Family of Gravitational Fields . Phys. Rev. 174, 1559 (1968), Cited 383 times

The covariant form of the metric tensor is expressed in terms of three parameters, m, e, and a by

ds² = ρ²dθ² – 2a sin²θdrdφ + 2drdu + …

Again, this is a line element, not a metric tensor.

___________________________

Marshall N. Rosenbluth, William M. MacDonald, and David L. Judd, Fokker-Planck Equation for an Inverse-Square Force. Phys. Rev. 107, 1 (1957), Cited 263 times.

Let the expression for distance between two points whose coordinates differ by dq1, dq2, and dq3 be

(ds)²= a_μνdq^μdq^ν,

Where a_μν is a metric tensor…

Again, a_μν is not a metric tensor, but a coefficient, when working in local coordinates from this (local coordinate specific) representation of the metric tensor: a_μν dx^μ⊗dx^ν…

___________________________

It isn’t that hard to say “line element”, or “matrix representation in local coordinates…”, or “matrix element in local coordinates…” instead of “metric tensor” or “metric” so why don’t we?

-S.C. Kavassalis

Originally on Blogspot here: http://sckavassalis.blogspot.com/2009/10/bad-language-metric-vs-metric-tensor-vs.htm

Physicists: Stop saying “Riemannian” when you mean “pseudo-Riemannian”. Yes, it does matter.

Some informal background: a Riemannian manifold is a differentiable manifold (where the tangent space at each point has an inner product) with a positive-definite metric tensor, d(x,y) ≥ 0.

A familiar Riemannian manifold is a Euclidean manifold (where one has to add a smoothly varying inner product on the tangent space of the standard Euclidean space), with the familiar Euclidean (distance) metric (our 3-space, for example).

What is NOT a Riemannian manifold is the familiar Lorentzian manifold of general relativity (of which the Minkowskian manifold of special relativity is a special case). The Lorentzian manifold is a pseudo-Riemannian manifold, the generalization of the Riemannian manifold, such that the metric tensor need not be positive-definite. This apparently seems like a minor point to some, but pseudo-Riemannian and Riemannian manifolds are incredibly different because of this.

One of the underlying assumptions of general relativity is that spacetime can be represented by a Lorentzian manifold with signature (+,-,-,-) or (-,+,+,+) – where the signature of a metric tensor is just the number of positive and negative eigenvalues of the corresponding real symmetric matrix once it is diagonalised.

Unlike a Riemannian manifold, with a positive-definite metric, a Lorentzian manifold M, with non-positive-definite metric, g, allows tangent vectors, X, to be classified into timelike g(X,X) > 0, null g(X,X) = 0, or spacelike g(X,X) < 0.

The causal structure of relativity comes from this classification.

Interestingly, when you most often are reading a paper in a physics journal though, instead of seeing “pseudo-Riemannian” you will see the word “Riemannian”; doing a search in the Physical Review Letters this afternoon for “Riemannian Manifold” yields 526 results, while searching for “pseudo-Riemannian Manifold” only yields 51. While I am sure a few of those authors were actually are working with Riemannian manifolds (and the obvious overlap with the “pseudo-Riemannian” search), the vast majority are simply misusing the term.

Some sample offenders:

Stephen A. Fulling, “Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time” (Phys. Rev. D 7, 2850 (1973), Cited 211 times) : Fulling technically means “pseudo-Riemannian space-time”, else he wouldn’t have any causal structure.

C. N. Yang, “Integral Formalism for Gauge Fields” (Phys. Rev. Lett. 33, 445 (1974), Cited 208 times). Yang starts a paragraph off with “Introduction of a Riemannian metric”, when he then must actually be introducting a pseudo-Riemannian metric. Later, when Yang is defining “Pure Spaces”, he says, “A Riemannian manifold for which the parallel-displacement gauge field is sourceless will be called a pure space.” He then asserts, “A four-dimensional Einstein space, ie. For which R_αβ = 0, is a pure space.” From the definition, if he really mean a Riemannian metric, he could not conclude that “a four-dimensional Einstein space” was a pure space, because an Einstein space must have a different signature to be causal (even though with R_αβ = 0 he is specifying that the metric tensor is locally isometric to a Euclidean space).

Almost anytime you see the phrase “Riemannian space-time”, they are being sloppy. There is no such thing as a Riemannian space-time.

All of these highly respected papers incorrectly refer to the spacetimes they are working in as Riemannian:

Friedrich W. Hehl, Paul von der Heyde, G. David Kerlick, and James M. Nester, “General relativity with spin and torsion: Foundations and prospects” (Rev. Mod. Phys. 48, 393 (1976), Cited 612 times)

David G. Boulware, “Quantum field theory in Schwarzschild and Rindler spaces” (Phys. Rev. D 11, 1404 (1975), Cited 117 times)

Kenneth Nordtvedt, “Equivalence Principle for Massive Bodies. II. Theory” (Phys. Rev. 169, 1017 (1968), Cited 88 times)

Leonard Parker and S. A. Fulling, “Quantized Matter Fields and the Avoidance of Singularities in General Relativity” (Phys. Rev. D 7, 2357 (1973), Cited 87 times)

M. J. Rebouças and J. Tiomno, “Homogeneity of Riemannian space-times of Gödel type” (Phys. Rev. D 28, 1251 (1983), Cited 65 times)

J. S. Dowker and Raymond Critchley, “Stress-tensor conformal anomaly for scalar, spinor, and vector fields” (Phys. Rev. D 16, 3390 (1977), Cited 59 times)

M. A. Melvin, “Dynamics of Cylindrical Electromagnetic Universes” (Phys. Rev. 139, B225 (1965), Cited 43 times)

Leonard Parker, “Conformal Energy-Momentum Tensor in Riemannian Space-Time” (Phys. Rev. D 7, 976 (1973), Cited 36 times)

A. A. Coley, N. Pelavas, and R. M. Zalaletdinov, “Cosmological Solutions in Macroscopic Gravity” (Phys. Rev. Lett. 95, 151102 (2005), Cited 32 times)

F. W. Hehl, E. A. Lord, and Y. Ne’eman, “Hypermomentum in hadron dynamics and in gravitation” (Phys. Rev. D 17, 428 (1978), Cited 20 times)

The list goes on, and on, and on…

Physicists (& Journal Editors): if you’re working in a causal spacetime (and you know you should be), don’t say “Riemannian”. Say, “Lorentzian”, or “pseudo-Riemannian”, or “non-Riemannian”, don’t be lazy.

You wouldn’t say “positive” when you mean “positive, zero, or negative“, so why would you say “Riemannian” when you mean “pseudo-Riemannian“?

-S.C. Kavassalis

Originally on Blogspot here: http://sckavassalis.blogspot.com/2009/10/bad-language-riemannian-manifold.html

Up first is Franklin Felber’s “Test of relativistic gravity for propulsion at the Large Hadron Collider” (available online: http://arxiv.org/abs/0910.1084)

My problem with this paper starts right with the second sentence of the introduction with this statement: “Within the weak-field approximation of general relativity, exact solutions have been derived for the gravitational field of a mass moving with arbitrary velocity and acceleration (Felber, 2005a).”

There are several points that should stick out in the mind of the reader. First, “weak-field approximation” and “exact solution” should not go in the same sentence. Perhaps, within the approximation it is exact, but it is not an exact solution (else it wouldn’t be an approximation). Second, “a mass moving with arbitrary velocity” is a pretty dangerous statement, because it suggests possibly ignoring the speed of light constraint.

Confusingly, when one follows the reference to the paper he is citing, “Weak ‘antigravity’ fields in general relativity”, we get another version of our initial point: “We recently derived and analyzed exact time-dependent field solutions of Einstein’s gravitational field equation for a spherical mass moving with arbitrarily high constant velocity”, where the ‘recent derivation’ in 2005 takes you to a paper from 2008 called, “Exact ‘antigravity-field’ solutions of Einstein’s equation”.

But back to the initial paper we are considering, and onto the third sentence: “The solutions indicated that a mass having a constant velocity greater than 3^-½ times the speed of light c gravitationally repels other masses at rest within a narrow cone.”

Totally ignoring the derivation of this result for the time being (which is not present in his paper or any of the initial citations), we will continue to analyze the language used here. The phrase “masses at rest” should stand out as odd to a relativist. Rest in terms of what, I wonder? Our arbitrarily fast, accelerating, mass? In what frame could the author possibly mean? “At rest” is a warning sign in any paper that claims to be written about relativity, because even basic students of special relativity should have the notion of ‘no absolute, well-defined state of rest’ drilled into them.

Fourth sentence: “At high Lorentz factors (γ >> 1), the force of repulsion in the forward direction is about -8γ⁵ times the Newtonian force.”

Again, simply looking at the language here, the phrase “Newtonian force” should jump out at you. What force are we talking about? In the Newtonian view of physics, we do refer to objects moving under the force of gravity, but in general relativity, we really should not. Gravity is simply a manifestation of the geometry of spacetime. An object moving along the curved spacetime manifold isn’t ‘moving under a force’, but rather, it is in inertial motion along a curved manifold. There is no force pushing objects out of straight paths, objects are still following the straightest path; gravity corresponds to the changes in the spacetime geometry along that path. Relativists should be careful not to ascribe a particle’s action to a ‘gravitational force’. While this is a pet peeve of mine, and a bad habit, good and respectable physicists do use the term “gravitational force”, partly out of habit, and partly because, in the Newtonian limit, it’s not so offensive.

Another quote to consider from the fourth sentence is, “in the forward direction”. Now our *Galilean* relativity should be telling us to be more precise with a statement like, but one can give the author the benefit of the doubt to assume he meant “forward” as along the path of our mass.

The second paragraph continually mentions this “exact-solution” to the Einstein equations, which is of course, just as dubious a claim as it was the first time the author made it. For those who aren’t familiar with the Einstein equations, they are non-linear PDEs that are quite difficult to solve exactly, which is why very few exact solutions exist (and they are all a big deal), and why most modern exact solutions are found numerically these days.

In the second paragraph, we have: “These exact ‘antigravity-field’ solutions were calculated from an exact metric first derived, but not analyzed, by (Hartle, Thorne and Price, 1986).”

Now, I am somewhat familiar with the reference he cites: “Black holes: The membrane paradigm”, edited by Thorne, Price, and MacDonald, but I am not familiar enough with the particular paper he citing, “Gravitational Interaction of a Black Hole with Distant Bodies” (by Hartle, Thorne, and Price) to know which “exact metric” he is referring to. Nevertheless, I do know that that particular paper was treating the “The long-term, secular evolution of a black hole weakly perturbed by gravitational forces of objects far from the event horizon is examined using the 3+1 formalism of the membrane paradigm”, which makes it fairly hard to guess what he would be referencing there.

It’s a little surprising that if Felber was actually working within the Membrane paradigm, that the word “membrane” doesn’t appear anywhere in the text of his paper, or “black hole”, or “event horizon”, for that matter. While often consequences derived from the study of event horizons are applicable in many other settings, it’s hard to see the connection the author is making in this case.

In Hartle, Thorne, and Price, an “exact analytical solution is found for the lapse, shift and spatial metric of a moving, nonrotating black hole” which leads Felber to claim, “The exact results confirm that a large mass moving faster than 3^-½c could serve as a driver to accelerate a much smaller payload from rest to a good fraction of the speed of light.” While I know I said I was just going to address the language here, I must point out that this claim/inference Felber is making seems quite without merit. He also doesn’t bother to assert how he has come to such a conclusion.

Onto the opening sentence of the third paragraph of the introduction: “The exact results are consistent with the repulsion of relativistic particles by a static Schwarzschild field, discovered
by (Hilbert, 1924).” Interestingly, his first attempt to back his claims up, outside of referencing himself or the strange appeal to Hartle, Thorne, and Price, comes in the form of a scan of section of Hilbert’s German version of his memoir, Die Grundlagen der Physik. Now, my German is pretty rusty, but Felber does cite a recent, English account of Hilbert’s curious result here: http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.1578v1.pdf.

Instead of going over the whole debate on gravitation repulsion, I’ll direct any curious reader to the paper that I believe should be the current authority on the topic. It is “Gravitational repulsion in the Schwarzschild field”, by McGruder (Phys. Rev. D 25, 3191 – 3194 (1982)). Very nicely, he goes over the historical background of the initial results of gravitational repulsion and the great number of papers that followed them. A small point to mention, Hilbert’s results, found independently by Bauer, were for particles near the Schwarzschild radius. Later, McVittie and Jaffe and Shapiro showed that repulsion could occur anywhere in the Schwarzschild field, so long as the total particle velocity was greater than 2^-½ c, not 3^-½c, like Felber is using.

Anyway, McGruder concludes an important result, which shouldn’t be a surprise these days to people who are familiar with similar solutions, that “gravitational repulsion can occur in the Schwarzschild field; but, it can only be detected by an observer whose meter sticks and clocks are not affected by gravity”. The important final line of his conclusion is, “that gravitational repulsion is not a function of the total particle velocity or energy; rather, its occurrence depends on the relationship between the transverse and radial velocity.” Unfortunately, it seems as if Felber is not familar with this work (ie. didn’t do a google search of “repulsive gravity”).

Now back to Felber: Nowhere near finished with the introduction, we have come to some fairly major issues. He is using a metric (although I see no evidence of him actually ‘using’ it anywhere), taken from the Membrane paradigm (not for a Schwarzschild field), using out of date results that only apply to near the Schwarzschild radius, and a very faulty interpretation of how these results can be interpreted/observed.

The ‘meat’ of the paper is his outline for an experiment to test his notion of gravitational repulsion at the LHC… so it can be assessed for the “potential of relativistic ‘antigravity’ for propulsion of payloads in the distant future.” Now, this claim seems so fanciful on it’s own, that many readers wouldn’t have bothered to give Felber a chance. Ruling something out, purely because it doesn’t fit with conventional knowledge is bad science. However, sloppy mathematics, ignoring current research, poor foundations, and leavings things as undefined as possible is also bad science.

-S.C. Kavassalis

Originally from Blogspot: http://sckavassalis.blogspot.com/2009/10/test-of-relativistic-gravity-for.html