# Quantum Moxie

thoughts of a selective subjectivist

## Spacetime: a more convincing argument

Following up on my previous post, here’s an even more convincing argument that it is possible to construct a toy universe in which the curvature of spacetime is due to electromagnetism and not gravity and that I was unjustly vilified over at nForum (remember while reading the rest of my post below that it was asserted that I lacked a rudimentary knowledge of general relativity).

Consider again a charged massless universe, that is a universe in which there are electromagnetic fields but no gravitational fields. In such a case the complete stress-energy tensor only includes the electromagnetic portion. Einstein’s field equations are

$G^{\alpha\beta}=8\pi T^{\alpha\beta}$.

In order to prove my point I need to show that $G$ is solely determined by $T$ (which I always thought was the standard interpretation, but after the tongue-lashing I received I’ll prove it just to be on the safe side).

In my toy universe I will assume that Riemannian geometry still exists (since, as a mathematical tool, it is independent of anything physical anyway). The definition of $G$ is

$G \equiv R^{\alpha\beta}-\frac{1}{2}g^{\alpha\beta}R$

where the $R$s may be written in terms of $g$. Thus it boils down to determining whether $g$, which is the metric, can be independently determined (i.e. from conditions not present in Einstein’s field equations).

As a simple case, let’s take the weak field approximation,

$g_{\alpha\beta} = \eta_{\alpha\beta} + h_{\alpha\beta}$

where $|h_{\alpha\beta}|<<1$ and $\eta_{\alpha\beta}$ is the Minkowski metric for flat spacetime. If we further restrict ourselves to the linearized theory, the linearized, weak field Einstein equations are

$\Box\bar{h}^{\mu\nu} = -16\pi T^{\mu\nu}$

where $\Box$ is the d'Alembertian. It is easier to explain the next step by showing what is done in normal GR, i.e. not in my toy universe. In such a case, for example,

$\bigtriangledown^{2}\bar{h}^{00}=-16\pi\rho$

and this is compared to the Newtonian

$\bigtriangledown^{2}\phi=4\pi\rho$

where $\phi$ is a scalar potential identified with Newtonian gravity. Thus, we choose

$\bar{h}^{00} = -4\phi$

in order to force this to match the Newtonian gravity!

Let’s switch back to my toy universe now. The energy density, $\rho$, is given by $T^{00}$ which, in normal GR is the density of the gravitational field. But in my toy universe,

$T^{00} = \frac{1}{2}(E^{2} + B^{2})$

where we have employed units such that $\mu_{0} = \epsilon_{0} = c = 1$. Further, in the electrostatic case, we note that,

$\bigtriangledown^{2}\phi = -\rho$

where $\rho$ is the charge density and where we again are employing units with $\mu_{0} = \epsilon_{0} = c = 1$. Thus in the electrostatic case of my toy universe we may choose

$\bar{h}^{00} = -16\pi\phi$

where $\phi$ is the charge density! The metric and thus the curvature of spacetime in my toy universe has absolutely nothing to do with gravity!

Addendum: If the full stress-energy tensor is employed in general relativity, i.e. with both gravitational and electromagnetic portions, one could presumably do a similar weak field construction in which $\bar{h}$ depends on both gravitational and electromagnetic fields which means I’m even right in this universe: the metric encodes the curvature of spacetime due to field sources and is not necessarily due solely to gravity.

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### 43 Responses to “Spacetime: a more convincing argument”

1. BlacGriffen Says:
April 17, 2010 at 6:06 pm

No good, your math looks fine but your conceptual framework is wrong. All you’ve shown is that you can get a space-time curvature, which is by definition gravity, from an energy density caused by an electromagnetic field. They are still entities which you could consider to be separate.

Put another way, you’ve demonstrated that photons couple to the gravitational field at the classical level even in the absence of anything else in the universe.

If you want to do this right start looking at QED, explicitly write in the U(1) gauge theory metric, and start making it dynamic. The next, though perhaps not necessary, step is to find a way of making the low energy effective Lagrangian linear in the field strength/curvature. If you’re up for a real challenge, though, you can consider a non-abelian gauge theory.

Even then, however, there’s still going to be one factor that separates gauge theories from gravity. And that is: the vector being parallel transported in the case of the gauge theories is purely orthogonal to the space-time manifold on which the theory is defined, whilst for gravity the vectors are tangent. So what in other theories is a clear distinction between Lorentz and group indices gets all mixed up in gravitational theories.

The other route, that I don’t know whether it’s been tried, would be to find a way to remove the dynamics from the metric in relativity. That one’s less of a tall order mathematically, since it would actually simplify the equations describing gravity, but more of an empirical problem to see whether or not you can explain all of the data (especially since you’d be basically throwing out big bang cosmology).

BG

2. quantummoxie Says:
April 17, 2010 at 6:25 pm

a space-time curvature, which is by definition gravity

And this is where I disagreed with the folks on nForum. It is precisely my point that this definition is arbitrary. Read the very beginning of Ch. 8 in Schutz’ book on GR (first edition). The logical premise that space-time curvature is gravity is axiomatic, i.e. it’s not derivable, it’s simply chosen. If you walk through the full derivation of Einstein’s field equations it is quite clear that this definition is a choice. In fact, it seems fairly clear given what I’ve shown above.

(By the way, you’re a mysterious figure, Mr./Ms. BlackGriffen. )

3. BlacGriffen Says:
April 17, 2010 at 6:58 pm

Heh. I’ve had this pseudonym since I was an undergrad, and I use it all over the place. My real name is Sean Lake, and I’m a grad student at UCLA. I don’t have anything published, yet, but I’ve got some stuff up my sleeve that I’m sure you’ll like.

As for gravity being space-time curvature I was under the impression that it is in an iff relationship with the gravitational/inertial mass equivalence principal coupled with the SR classification of inertial mass as a type of energy. Certainly you can imagine a universe in which all particles have a fixed charge to rest mass ratio, including the force carriers, and in that case it would likely not be possible to cleanly separate the effects of one force from another. In fact that is hinted at by the inquiries in the previous post about the use of different objects to probe the field. You can find a similar discussion in Jackson in the section where he discusses the presence/absence of magnetic monopoles. He shows that it is possible to mix the electric and magnetic fields in such a way that the correct question is not, “Why are there no monopoles,” but, “Why is the electric charge/magnetic charge ratio the same for all particles?”

Speaking of magnetic monopoles, I had an interesting experience during one journal club a couple of years back. We were getting into a discussion of topological solitons and part of the lead-up required a discussion of monopoles. I’ve never been a fan of the Dirac string since it throws out the property of the vector potential used to justify B = curl(A) and E = – grad(phi) – diff(A, t) in the first place (continuous/differentiable). So, I introduced another vector potential (can’t recall what letter I gave it at the time, so I’ll use ~A) which corresponded to another gauge symmetry. Specifically, if I recall correctly, the symmetry that produced the correct properties was a chiral U(1) (ie exp( i q_b * gamma_5 * phi(x) ) ). It had all the right properties, and I even tried to gauge the electric/magnetic charge rotations since that would immediately rule out being able to observe such a difference. Needless to say, that effort failed but I hadn’t completed it in time for the presentation. It did have the correct rotation properties between q and q_b, though, so I was happy with it. I wonder if something interesting could be said by combining this treatment with a full QFT treatment including the fact that the global chiral symmetry is “anomalously” broken by triangle diagrams. Well, the chiral symmetry isn’t necessarily broken so much as we have to choose between breaking U(1) chiral and U(1), so we keep the U(1).

After the presentation I was politely informed that all of the monopole dynamics were typically accounted for by using the ordinary vector potential and allowing for Dirac strings. Nobody mentioned whether anyone had bothered to look at it in the light that I had, though.

So I still think that a Dirac string monopole is something that we’ll never observe, but we’ll see.

4. quantummoxie Says:
April 17, 2010 at 7:14 pm

Aha! But by revealing your true identity you have now taken all the mystery out of it. Either way, publications or not, you clearly know what you’re talking about so kudos. If you’re interested in some deeper details, both Joe Fitzsimons and I are exploring a bit of this offline.

As for gravity being space-time curvature I was under the impression that it is in an iff relationship with the gravitational/inertial mass equivalence principal coupled with the SR classification of inertial mass as a type of energy.

Right, but, again that’s an axiomatic assumption about the nature of space-time curvature. Space-time curvature is described by a metric, but a metric is just a mathematical thing. Mathematicians (as is their wont) have completely generalized metrics.

Let me put it this way: my assertion does not “break” the equivalence principle nor does it “break” the notion that inertial mass is a form of energy. That’s still as true as ever.

5. BlacGriffen Says:
April 17, 2010 at 8:09 pm

Thanks for the offer, but I’ve got a pretty full plate as it is.
Right, but, again that’s an axiomatic assumption about the nature of space-time curvature. Space-time curvature is described by a metric, but a metric is just a mathematical thing. Mathematicians (as is their wont) have completely generalized metrics.
Well, that skips a step. The connection is what defines the curvature. Making the metric dynamical is a step that’s always taken but I haven’t seen a convincing argument for why the connection has to be metric compatible and thus why the theory should be defined in terms of a dynamic metric instead of a connection. Maybe I missed that day in GR.

As for the axiomatic assumptionness of it, I think there’s a pretty good case for dividing forces between those that arise due to parallel transport of vectors tangent to the manifold (ie with indices that correspond to directions of travel along the manifold) and those that are orthogonal (ie elements of a vector space that has no a priori connection to the manifold). Now you may argue that I’m artificially dividing a larger manifold into space-time and “other,” and that would be a fair criticism. I don’t actually have a good reason for doing so, but it is standard practice. Consider, for example, the functional integral form of QFT. The n-point functions are all of the basic form:

integral( D[field strength] * (field strength factors) * exp(i integral( d(spacetime) Lagrangian density )))

So in some sense you’re integrating along part of the extended manifold outside of the exponential and part inside. I understand that it works, but am kind of curious if there’s an explanation for why. Or, equivalently, if there’s a way to unify everything onto a single manifold with a single unified treatment for the variables that describe the configuration of, for lack of a better term, reality where all of the parameters are treated on an equal footing.

BG

6. BlacGriffen Says:
April 17, 2010 at 8:10 pm

D’oh! Using HTML like quote tags doesn’t work. I’ll have to try something else next time…

7. quantummoxie Says:
April 17, 2010 at 9:51 pm

For including LaTeX, embed the LaTeX code inbetween \$latex on the left and a single dollar sign on the right.

I need to hit the sack, as they say, so a fuller response will have to wait until tomorrow, but here’s a quick response. You’re absolutely right that what you’re saying is the standard interpretation. My point is that we’ve abstracted almost beyond meaning in some cases (especially in certain aspects of QFT). So, for instance, there are certain things that are taught in QM in a certain way that, if you start asking questions, hinge on whether or not Hilbert spaces are real or not. People either brush this aside or ignore the contradictions. To me, as an empiricist, there are similar problems in interpretations of GR.

As for the connection defining the curvature, that’s an interesting point. Let me sleep on it.

8. Dear Ian,

In 1925, G.Y. Rainich provided the necessary and sufficient conditions for a Lorentz spacetime to be solution of Einstein-Maxwell equations. One account, simplified by the use of spinors, can be found in “Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields (Cambridge Monographs on Mathematical Physics)”. There are three algebraic conditions and one differential. These conditions were rediscovered by Misner and Wheeler in 1957, and are presented e.g. in “Gravitation – An Introduction To Current Research”, 1962, ed. Louis Witten, chapter 9. In addition to rediscovering the condition, they showed that a pair electron-positron can be interpreted, based on cohomology, as the two mouths of a wormhole – the electric field lines don’t need to originate and end in a source – they can be with no beginning and no end (chapter 10, see also “Gauge Fields, Knots and Gravity” by John Baez and Javier Muniain).

Rainich’s algebraic conditions for the energy-momentum tensor to be sourced by the electromagnetic field are

$A1. T^i{}_i=0$

$A2. T^i{}_k T^k{}_j \cong \delta^i{}_j$

$A3. T_{ij}v^iv^j\geq 0$ for any timelike vector $v^i$

Rainich’s differential condition:

$D0. \nabla_i S_j=\nabla_j S_i,$

where

$S_j:=\frac {\epsilon_{ij}{}^{kl}T_{mk}\nabla_l T^{im}} {T_{rs}T^{rs}}$

If these conditions are satisfied, there is an electromagnetic field which can source it, uniquely determined up to a duality rotation – a complex rotation where the complex structure is given by the Hodge * operator acting on 2-forms.

Is it possible for other sources of the gravity field to respect Rainich’s conditions? Take for example a fluid in thermodynamic equilibrium, characterized by the density $\rho$, pressure $p$, and 4-velocity u. It’s energy-momentum tensor is

$T^{ij} = (\rho + \frac {p} {c^2}) u^i u^j + p g^{ij}$

In diagonal form, for the metric signature (-+++), the energy-momentum tensor looks like

$T^i{}_j = diag(-c^2\rho,p,p,p)$

The Rainich algebraic condition A2 is satisfied because of the diagonal form, and the energy condition A3 is satisfied too. In order to be trace-free (condition A1), it has to satisfy

$3p = c^2 \rho.$

The difficult problem seems to be to find solutions which satisfy the differential condition. How can the fields $\rho$, p and u be taken so that both Einstein’s equation, the condition above, and the condition D0 be satisfied?

Cheers,
Cristi

9. Jose Ignacio Says:
April 18, 2010 at 7:58 am

The statement:

“The energy density, $\rho$ , is given by $T^{00}$ which, in normal GR is the density of the gravitational field”

is false, in normal GR $T^{\mu\nu}$ is the energy-momentum tensor of matter (or fields other than gravity). In normal GR the gravitational field has not definite energy-momentum tensor because gravity is replaced by geometry, and this is a valid replacement because the equivalence principle.

10. quantummoxie Says:
April 18, 2010 at 11:12 am

Cristi,

Thanks for the reference. I will need to find it and read it.

Jose,

My words were poorly chosen there. What I mean was that the associated energy density in GR is the energy density of the gravitational field which is exactly what you’re saying.

My point (and the work by Rainich seems to support this interpretation) is that you can interpret the geometry of spacetime differently without breaking the equivalence principle.

Ian

11. Dear Ian,

You’re welcome.

I think that it should be plausible that the same geometry of spacetime can be obtained from different fields, knowing that the matter fields have, in total, more degrees of freedom than the Ricci tensor. Even in the solution of Rainich, the same curvature can be obtained from different electromagnetic fields, obtained from one another by a duality rotation. In the case of the wormhole “charge without charge” solution of Wheeler and Misner, this ambiguity is removed by requiring the source to be electric charge only, not magnetic monopole or combination.

In the case of the fluid, I checked only the algebraic conditions, and not the differential one. The differential condition translates in a PDE on the fluid’s flow satisfying the constraint $3p(t,x) = c^2 \rho(t,x)$, but I did not investigated whether it has solution. Maybe it is an interesting research to see what other matter fields curve the spacetime like the electromagnetic field does.

The main point is that there must be some matter to mimic the energy of an electromagnetic field, because in vacuum the energy-momentum vanishes, and the geometry of a vanishing energy-momentum is different from that of a non-vanishing one. Thinking at an example with a box in vacuum, the geometry outside the box is very different if the box contains a massless charge, than if it contains a neutral massive object. Outside the box, if the source is the mass only, the spacetime is Einstein, being a solution to Einstein’s vacuum equation. If inside the box there is a charge, even a massless one, outside the box cannot be, in fact, vacuum, because of the electromagnetic field’s energy. This is why I interpreted your affirmation “The energy density, $\rho$, is given by $T^{00}$ which, in normal GR is the density of the gravitational field” as referring, in fact, to the energy density of a gravitational source, possibly a material fluid, not the gravitational field itself.

Best regards,
Cristi

12. [...] In the responses to my previous post, Cristi Stoica provided some very useful references that, for now, I am going to classify as “Rainich unification” since I needed something [...]

13. Tim van Beek Says:
April 21, 2010 at 8:48 am

Hi there,

let me start with some (hopefully) simple question about nomenclature: Ian, you write that in your toy universe there are no gravitational fields, but only electro-magnetic fields.

I am used to gravity = curvature of spacetime = spacetime metric = gravitational field.
So “graviational field” has to mean something else, right?

You write: “In order to prove my point, I have to show that G is solely determined by T”.

The field equations are usually seen as the only equations that determine G and T (and if the folks at the nForum said otherwise that has escaped me), so I guess the problem is more about what the interpretation or the content of G and T are?

14. quantummoxie Says:
April 21, 2010 at 11:55 am

Tim,

Yeah, the standard interpretation is that curvature = gravity which one could still insist on, but it would then be an argument over semantics.

1. Yakir Aharonov’s interpretation of a version of the Aharonov-Bohm effect is that the shifting of the electron’s interference pattern in said effect is due to the topology of space (Aharonov & Rohrlich, p.49). If the curvature (topology) of space is just gravity, there must be some gravitational effect going on here. Why hasn’t anyone considered it?

2. Let’s take the standard interpretation that gravity is the curvature of spacetime. Now we know from QED (actually from simple relativity, really – mass = energy) that photons (E&M fields) can produce curvature. Urs says this is simply gravity (to which I say, fine, but it’s a semantic thing). But if this is true, suppose we have a phenomenal concentration of photons in one location such that spacetime is warped enough that it’s a macroscopic effect. You should, in some way, be able to recover Newtonian gravity from this situation if the curvature is simply gravitational. But will we? And does it hold up experimentally (admitted this is hard to test)?

Pure speculation: I have a hunch there’s a greater symmetry (duality?) between electromagnetism and gravity than people are willing to admit. Even Rainich questions this (in fact that is the point of his paper).

15. Ian,

1. The Aharonov-Bohm effect is a topological effect, indeed. Only it is related to the topology of the gauge bundle of the electromagnetic field, rather than that of the spacetime itself.

16. BlackGriffen Says:
April 21, 2010 at 2:36 pm

Are you looking to re-invent Kaluza-Klein theory here? http://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory

17. quantummoxie Says:
April 21, 2010 at 2:57 pm

Cristi: as an empiricist, my first question is, what physical significance does the gauge bundle have? Spacetime has some kind of physical meaning and interpretation. I’m not sure gauge bundles do.

Sean: LOL, God no. My PhD dissertation was on Eddington’s Fundamental Theory which built heavily on Kaluza-Klein theory. I’ve seen enough of that stuff. It’s much closer to Wheeler’s geometrodynamics, but taking a step back and trying to “reconcile” the “language” as it were. I know that sounds vague, but I have to run to a meeting.

18. Tim van Beek Says:
April 21, 2010 at 3:37 pm

I’ll try if I can use some tags here…

Ian wrote:

…the standard interpretation is that curvature = gravity which one could still insist on, but it would then be an argument over semantics.

I guess we agree upon the meaning of curvature as a mathematical object, living on a smooth manifold. But then what is gravity to you?

If the curvature (topology) of space is just gravity, there must be some gravitational effect going on here…

Like Cristi Stoica said:
The mathematical model is R^3 with the z-axis removed and the electrons moving in the x-y-plane (you may add a time dimension as you wish), the point is: the curvature is zero everywhere, ergo (using my nomenclature) no gravity. The effect is explained by the fact that the manifold the electrons move in is not simply connected, i.e. has a nontrivial fundamental group. Same thing as with Dirac’s monopole, that’s a purely topological effect in a flat space, too.

Ian wrote:

…to which I say, fine, but it’s a semantic thing…

Perhaps if you try to rephrase what you mean here in one or two different ways I will stand a better chance to follow you.

Ian wrote:

suppose we have a phenomenal concentration of photons in one location such that spacetime is warped enough that it’s a macroscopic effect.

Theoretically that’s the case – from the viewpoint of an external observer at infinity – when looking at the event horizon of a (Schwartzschild) black hole that sucks in a nearby star, plus the black whole could have been created – theoretically – by the gravitational collapse of a bunch of photons. My point being: I don’t see anything special or new or paradox in this Gedankenexperiment.

You should, in some way, be able to recover Newtonian gravity from this situation…

That’s possible in the weak field approximation only, and – again – if you like to calculate the gravitational field of a dense bunch of photons, go ahead, I don’t get the point where there is something mysterious (that’s not critisism, I’m trying to indicate where I’m stuck in following you).

Ian wrote:

Pure speculation: I have a hunch there’s a greater symmetry (duality?) between electromagnetism and gravity than people are willing to admit.

If we are talking about a unification of gravitation and electromagnetism as classical field theories, then this has of course a long long history (see the review I linked to over at the nForum).

Ian wrote:

Spacetime has some kind of physical meaning and interpretation. I’m not sure gauge bundles do.

The gauge bundle keeps track of the phases of (localized) excitations of (quantum) fields, as a mathematical device it allows to calculate e.g. interference patterns. I suppose you know the cute little book “QED: The Strange Theory of Light and Matter” by Feynman and his clock-analogy of the phase of an electron?

19. quantummoxie Says:
April 21, 2010 at 9:13 pm

Couple of clarifying points:

I agree that there really isn’t anything remarkable here. What I’m arguing for is consistency in thinking and rationalizing definitions. It’s very, very similar to arguments I made in my FQXi essay and similar to a general philosophy espoused by Tom Moore. It’s born out of the desire to explain physics to the undergraduate non-majors that I teach without oversimplifying it.

Regarding the Aharonov-Bohm effect, see my comment below that I hope will clarify things (but you never know since I’m watching hockey at the moment). Ditto that for fiber bundles and semantics. I don’t know what I was thinking in regard to the ball of photons.

Anyway, here’s another attempt at explaining my position:

Spacetime clearly is some kind of physical “surface” that can be modeled mathematically by a manifold. This manifold clearly also includes areas of local curvature near large concentrations of mass/energy. Thus the standard interpretation is that the curvature of spacetime is gravity. I actually have no objection to this statement.

But the standard interpretation goes further and says the curvature of spacetime is and only is gravity, i.e. it defines gravity to be the curvature of spacetime. That’s the part I have a problem with (hence, I agree with David – let’s just dispose of the word entirely).

Here’s why I object. First, the sole motivation for this is historical because that was what Einstein was attempting to figure out at the time. Second, it is my contention that charge ought to do the same thing independent of its association with any mass. Now the usual response to this is that I’m automatically associating gravity with mass forgetting that gravity can be associated with pure energy since mass and energy are equivalent. This is the motivation for saying that energy causes curvature. But, the only self-consistent definition of mass, i.e. that makes sense in both single and multi-particle systems, is as the magnitude of the four-momentum vector (Tom Moore has an excellent discussion of this in his textbook). This implies, if you follow the logic all the way through, that while mass is energy, energy isn’t necessarily always mass. Thus, since the word “mass” was historically associated with “gravity” thanks to Newton, the assumption is that all of energy should be as well.

I find this reasoning to be somewhat circular particularly when you start viewing things from the standpoint of the so-called “interaction picture” envisioned by the ideal that the Standard Model strives for since it would imply that photons and other massless particles could exchange gravitons. But that throws all sorts of wrenches into the particle interpretation which has nice and neat classifications for particles based on their characteristics (mass, charge, spin, etc.). Nevertheless, what we’re arguing about here is a word, not the mathematics or even how to use it. Hence it’s really just semantics.

As for fiber bundles, this relates again to my FQXi essay. You say,

The gauge bundle keeps track of the phases of (localized) excitations of (quantum) fields…

That tells me what the bundle “keeps track of.” It doesn’t tell me what the bundle is physically. This is part of what I tend to think of as the “over-abstractification” of physics. When does it stop being physics and simply become math? One never knows when this stuff could actually make its way into technology. In fact quantum gravity might be more important technologically than people think – gravity is known to degrade entanglement and they’re currently trying to develop satellite-based entanglement systems. Suppose we actually reach the point where this stuff matters technologically. Spacetime makes intuitive sense to most engineers. It’s “substantive.” You can work with it. How do you work with a fiber bundle?

20. BlackGriffen Says:
April 21, 2010 at 10:52 pm

Yeah, it gets tricky when you define mass as $p_\mu g^{\mu\nu}p_\nu$ and you start to let the metric vary, but what else is new? I mean, last I heard there isn’t even a single agreed upon definition of energy in GR, so why should the mass picture be any clearer?

On the subject of mass, there’s a fun example in E&M where the photons behave like massive particles. I’m not talking about the difficult picture of superconductivity, I’m talking about sending signals down a wave guide or coaxial cable. When you do that you get a dispersion relation that looks exactly like the equation for mass/energy relation in SR, modulo units of course.

BG

21. Ian,

you said “as an empiricist, my first question is, what physical significance does the gauge bundle have? Spacetime has some kind of physical meaning and interpretation. I’m not sure gauge bundles do.”

The empirical component of the gauge bundle is given by the Aharonov-Bohm effect itself. It is an effect of the holonomy of a connection (which is the electromagnetic potential) on the U(1)-bundle.

To avoid loading your page too much with my own viewpoint, and to have more control of the equations, I provide more details here: http://www.unitaryflow.com/2010/04/vec-bund-fundam-physics.html

22. Tim van Beek Says:
April 22, 2010 at 7:23 am

I have some minor remarks below, but the main point, I think, is this (don’t know if it is entirely clear, but I hope it let’s us advance a bit):

In GR space and time do not have any “reality” on their own, they are defined as causal relationships with respect to the gravitational field (I know, I should not use this term, but did not come up with something better). You cannot define a universe without any effects related to gravity. If you visualize a manifold as spacetime and then put fields on it, the gravitational field being one of them, then this picture is misleading.
If you put an electromagnetic field on Minkowski space, you are implicitly saying “there is gravitation=curvature and is is zero everywhere and the energy of the electromagnetic field is too weak to couple to it”.

Example (very famous, has a name that I don’t remember): Fill a bucket with water. The water is still. Now let the bucket rotate. Let the water rotate with the bucket, now the surface is curved. How does the water “know” that it is in motion and has to form a curved surface? The answer of GR is: It is in motion with respect to the local gravitational field. And that is there, even when we are in Minkoswki space. But being in Minkowski space enables you to tell when you are not in an inertial system, by comparing your motion to the local gravitational field.

Ian wrote:

But the standard interpretation goes further and says the curvature of spacetime is and only is gravity, i.e. it defines gravity to be the curvature of spacetime. That’s the part I have a problem with (hence, I agree with David – let’s just dispose of the word entirely).

I’ve no problem with disposing with the term gravity, but you seem to see a crucial difference between the curvature of spacetime and gravity, and that is a reason not to dispose with it, isn’t it?

Ian wrote:

Here’s why I object. First, the sole motivation for this is historical because that was what Einstein was attempting to figure out at the time.

That’s not really an argument against anything, unless you would like to stress that some label is a historical accident due to a misprint, or that we should rename general relativity because all students keep asking which side of the civil war he fought for; so as an advisor in rhetoric I’d recommend to drop that point entirely

Ian wrote:

Tom Moore has an excellent discussion of this in his textbook.

I don’t know that book, could you provide a reference?

Ian wrote:

…while mass is energy, energy isn’t necessarily always mass.

The standard interpretation is that mass is one particular form of energy, and that vice versa all forms of energy couple to gravity. I know that there is some more important point here that you try to make, but – again in my role as rhetoric advisor – most physicist will understand this as stressing the perfectly obvious

Ian wrote:

But that throws all sorts of wrenches into the particle interpretation…

I’m no expert in string theory but string theorists see no problem here (as didn’t Feynman when he reinvented the graviton in his lectures about gravitation), could you elaborate on that?

Ian wrote:

Spacetime makes intuitive sense to most engineers. It’s “substantive.” You can work with it. How do you work with a fiber bundle?

Only if we are talking about the concepts of time and space built from everyday intuition. One lession of 20th century physics is that nature is essentially different in certain aspects from our everyday intuition. In GR the question “what happens now on Alpha centauri?” is meaningless, because there is no objective meaning to “now”. Given that, spacetime defines causal relationships of events. Gauge bundles add to that description. Neither of them is really close to what our brain learns from everyday input.

23. quantummoxie Says:
April 22, 2010 at 8:49 am

Let me just lead off by saying, you all haven’t said anything I’m not completely aware of. I know I’m bucking a commonly held view here and everyone is free to disagree with me. But my should be respected. It should be obvious by now that it is not born out of ignorance but out of a desire to reinterpret certain cherished ideas (which is why, I suppose, it gets attacked).

If you visualize a manifold as spacetime and then put fields on it, the gravitational field being one of them, then this picture is misleading.

I agree. The manifold should be the field. To quote Rainich,

…when the space is given all the gravitational features are determined; on the contrary, it seemed that the electromagnetic tensor is superposed on the space, that it is something external with respect to the space, that after space is given[,] the electromagnetic tensor can be given in different ways.

This was historically unsettling to many folks who tried to find ways around it: Weyl, Eddington, Rainich, Kaluza, Klein, et. al. But this philosophical approach to unification has waned, largely because most people think it was a failure – which it was, but, personally, I think it was a failure due to a lack of knowledge rather than a poor base model/idea. So much advancement has occurred since those theories were first considered that I think revisiting the base idea might be worthwhile (ok, so maybe I lied about Kaluza-Klein theory, though my approach is entirely different since I haven’t gone as far as postulating additional dimensions).

I’ve no problem with disposing with the term gravity, but you seem to see a crucial difference between the curvature of spacetime and gravity, and that is a reason not to dispose with it, isn’t it?

I guess I’d agree to keep it if it were agreed that gravity didn’t define spacetim curvature (curvature clearly defines gravity, but I think it should be open to interpreting spacetime curvature as being a more general notion).

That’s not really an argument against anything,

Logic compels me to disagree. Prior to 1905, time was treated as being absolute. Why? Because, historically, that’s what Newton did and no one thought to question it until relativity appeared on the scene.

most physicist will understand this as stressing the perfectly obvious

I’m a bit confused about which point you’re referring to here. Because I know a number of physicists who are sympathetic to my view.

One lession of 20th century physics is that nature is essentially different in certain aspects from our everyday intuition. In GR the question “what happens now on Alpha centauri?” is meaningless, because there is no objective meaning to “now”.

I both agree and disagree. Yes, relying on our intuition is not the safest course of action as 20th century physics has taught us. But there is still room for rational, well-formulated experiment/observation and, like it or not, science will never be completely objective (good luck to Harry on trying to fix that). So completely ignoring our everyday experience is also not a wise idea. There needs to be a balance. Where that balance point is, is what we’re debating. So, suppose you were Newton but you stumbled onto essentially the same observations that Eddington did in 1919. It seems to me that the warping of at least space (not time if you’re Newton) is something even Newton would have arrived at eventually given that kind of observational evidence. I can’t say the same about gauge bundles.

24. quantummoxie Says:
April 22, 2010 at 8:53 am

Oh, and, while it is an introductory textbook, Moore’s book espouses an entirely unique philosophy (in my opinion) and is thus worth reading solely for that (I use it to teach intro classes):

Six Ideas That Shaped Physics, by Thomas A. Moore

Comes in six thin volumes: Units C, N, R, E, Q, and T. While I don’t agree with everything he says, I agree with most of it. His motivating principle for writing it (and it was a 10-year project) was to explain things correctly the first time so that it wasn’t necessary to go back and say, “well, we lied to you in intro physics.” He is a frequent collaborator of Dan Schroeder’s. Dan wrote a phenomenal text called Introduction to Thermal Physics which is based on similar guiding principles. Dan also co-authored a QFT text with Peskin.

25. Tim van Beek Says:
April 22, 2010 at 10:46 am

Thanx for the reference to Moore, I will have a look at it.

…curvature clearly defines gravity, but I think it should be open to interpreting spacetime curvature as being a more general notion…

My problem is that I understand “define” in a mathematical sense, and that means both notions are equal. And I think the same happend over at the nForum. Maybe an example of what you have in mind for the notion of gravity would help (sorry, but I still did not get that).

Ian wrote:

I’m a bit confused about which point you’re referring to here

To the statement “matter is always energy, but energy is not always matter”.

…even Newton would have arrived at eventually given that kind of observational evidence. I can’t say the same about gauge bundles.

Ok, one should not completly dismiss everyday intuition, gauge bundles are far removed from that, farther than the notion of spacetime, and maybe one should try to remedy that. I think we can agree on that (but I suspect that some folks over at the nForum would dismiss this kind of reasoning as utterly irrelevant, but then not everybody has to teach non-major undergraduates .

I think I understand your viewpoint a little bit better now (modulo my first question above), did I succeed in explaining what troubled some people about your statements over at the nForum?

26. Tim van Beek Says:
April 22, 2010 at 12:55 pm

I wrote:

Maybe an example of what you have in mind for the notion of gravity would help…

A telltaling mistake, I thought you meant gravity should be open to a more general interpretation. About spacetime curvature I think: Well, it defines locally what an inertial system is and if you are not in one then you will experience forces that we know from everyday live as gravity. Being in an inertial system means free falling means movement along a geodesic means gravitational force is zero etc.etc.

I have a hard time imagining that curvature could have some additional role that is not connected to this set of ideas and rephrase my former question to refer to curvature now. Is your idea that curvature should in addition model electromagnetic forces?

(I hope that my questions provide an indication what kind of introduction you could formulate for your ideas, to avoid misunderstandings, if you talk to people who think like me

27. BlackGriffen Says:
April 22, 2010 at 2:45 pm

Gravity and spacetime curvature are in principal separate concepts for one very basic reason: we can easily imagine a universe in which the spacetime manifold were rigid and thus did not curve in the presence of the stress energy tensor. The force we identify as gravity would then be an ordinary force that has the macroscopic appearance of the manifold flexing due to stress-energy, but that is independent of the manifold microscopically. Please correct me if I’m wrong because this is what I understand the string theory interpretation of gravity to be.

Toward that end we also still perform tests to look for a difference between inertial and gravitational mass. I don’t recall to what level they’ve been verified to be the same, but like tests for a photon mass just about everyone expects the results to be negative.

28. Aha! Excellent points Sean! And what about all those nutty 5-D gravity folks who were involved with Gravity Probe-B? I’m pretty sure they see gravity and spacetime curvature as separate.

I think I understand your viewpoint a little bit better now (modulo my first question above),

LOL, I love the modulo reference. Total math geek, you are. :p

but I suspect that some folks over at the nForum would dismiss this kind of reasoning as utterly irrelevant, but then not everybody has to teach non-major undergraduates

But that’s not the only thing that motivates me. Any description of the universe ought to be both consistent and reasonable (there are better words to use here but I can’t remember them – there’s a whole stack of writing about this, in fact, by people who work in foundations). I work in foundations most of the time and so I ask questions that are unpopular, subtle, difficult, and supposedly resolved because there are always overlooked nuggets of useful ideas floating around out there just waiting to be discovered (or re-discovered in many instances).

My problem is that I understand “define” in a mathematical sense

Bad choice of word on my part, though that is the word that is generally used. I would have used some mathematics-like analogy to sets and subsets or something, but, knowing mathematicians I would have been picked apart on semantics when it was really a bigger picture concept I was after.

did I succeed in explaining what troubled some people about your statements over at the nForum?

I think what you did is drive home a point I already knew (I had a similar bad experience at MathOverflow). I just couldn’t find a way to properly formulate my point (I have that problem sometimes in discussiony situations).

29. Tim van Beek Says:
April 23, 2010 at 2:53 am

Ian wrote:

I think what you did is drive home a point I already knew (I had a similar bad experience at MathOverflow). I just couldn’t find a way to properly formulate my point…

That’s what I meant by “cultural clash”. If you visit a different continent you know there will be some problems due to misunderstandings. The problem here is that we encounter a cultural difference when we do not expect to see one, and one that is not easy to spot, because we all use “the same language” (by which I mean both English, math etc.).

So what I try to do here is explain what I am thinking and why I am mislead by (the specific formulation of) certain statements you make, I do not try to make any specific points about some interpretation of GR

Total math geek, you are.

I’ll take that as a compliment
But seriously, that’s the way of talking people over at mathOverflow and the nForum are used to, and if you would like to be understood you have to adapt to a certain degree.

BlackGriffen wrote:

The force we identify as gravity would then be an ordinary force that has the macroscopic appearance of the manifold flexing due to stress-energy, but that is independent of the manifold microscopically. Please correct me if I’m wrong because this is what I understand the string theory interpretation of gravity to be.

I can’t speak for the whole string community and maybe there are some who would agree with you, the interpretation that I know of is this: In string theory you fix a spacetime as a background, the objects of string theory generate gravitons. The situation of having a (classical) spacetime with gravitons on it is a quantum perturbation of a classical background.

String theory does not change anything in the interpretation of classical GR, it just tries to incorporate quantum gravity effects as a perturbation of the classical situation. One critisism of string theory therefore is that one cannot know if perturbation theory is applicable. Example: If you have a gravitational collapse in a classical situation, can this be modelled by increasing the number of gravitions when starting from a flat spacetime?
Both sides (stringy and antistringy) agree that this is an open problem, they disagree about the importance of it.

• BlackGriffen Says:
April 23, 2010 at 3:09 am

Tim wrote:
[quote]I can’t speak for the whole string community and maybe there are some who would agree with you,[/QUOTE]
I can’t speak for any of them. I just don’t see how the collective action of wiggling strings alters the geometry of the spacetime manifold, or the target space. Unless space itself is somehow constructed out of these strings, of course, but I never delved deep enough to figure out what string theorists interpret these strings to be. If you start trying to reduce space to some kind of web of 1-dimensional objects (or 1+1 dimensional, depending on how you look at it), then it begins to sound vaguely loop quantum gravity-esque to my essentially outsider’s perspective. In such a picture, perhaps, the strings could be the edge of the simplexes that the LQG people use to construct the normal 4-dimensional spacetime.

in all seriousness, what are the strings? What is it that is supposed to be vibrating?

Thanks,
BG

• Tim van Beek Says:
April 23, 2010 at 9:30 pm

BlackGriffen wrote:

…in all seriousness, what are the strings? What is it that is supposed to be vibrating?

That is the input to the theory, there is no further explanation of what a “string” is. Looking for one will be in vain.

30. quantummoxie Says:
April 23, 2010 at 7:08 pm

I actually had thought that the Rainich conditions led to geometrodynamics which in turn gave birth to (among other things) loop quantum gravity. Did someone say that in this discussion thread? I can’t remember and I’m too tired to check.

Sean’s points about strings are one of the main reasons I’ve always had issues with string theory. I was enamored with them for awhile in the ’90s and nearly went into it at one point, but the empiricist in me started to ask questions and I didn’t find the canned answers to be all that satisfying.

Well, anyway, I still think everything seemed to be going fine until Urs went postal on me. And Urs is supposedly a physicist so he should have understood what I was talking about.

Look, I don’t preclude that I am wrong. I just think it was a point worth discussing rather than not only dismissing it but getting personal about it (though that, I think, was born out of something unrelated that didn’t occur to me until today and I’ll leave it at that).

31. Tim van Beek Says:
April 23, 2010 at 9:43 pm

Well, anyway, I still think everything seemed to be going fine until Urs went postal on me.

No. The discussion if “Lorentzian spacetime” is the correct term for manifolds that are candidates for spacetimes set the stage. That was the first step into the wrong direction.

And Urs is supposedly a physicist so he should have understood what I was talking about.

My whole reason for being here is to point out that one cannot conclude that “x is a physicist, therefore he should understand that…”
That is not true. You have to anticipate that there will be misunderstandings. Its unavoidable. Its nooone’s fault. It happens. You have to understand this or you will never overcome them. I was not successful to explain that, but I hope someone else will be.

A dieux.

32. quantummoxie Says:
April 24, 2010 at 8:46 am

What I meant was that the tone of the conversation changed when Urs got fed up. Yes, you’re absolutely right, I should anticipate that there will be misunderstandings. I wrote the above comment after a long and tiring week. My point was that I’m still mystified as to why a discussion about a scientific point has to turn personal. Why not simply say, “OK, majority rules, it will be X” rather than adding “you’re a crappy physicist, go away.” I would have simply shrugged my shoulders and shut up.

• phorgyphynance Says:
April 27, 2010 at 2:42 am

Where did anyone ever say you were a crappy physicist? Harry doesn’t count

33. Steve Dufourny Says:
April 26, 2010 at 6:54 am

Hi all, Cristi, Ian,

Very beautiful blog and of course discussions.

Best Regards

Steve

• quantummoxie Says:
April 26, 2010 at 7:48 pm

Hi Steve! Thanks for the comment!

• Steve Dufourny Says:
June 1, 2010 at 4:53 am

You are welcome, with pleasure.

34. phorgyphynance Says:
April 27, 2010 at 2:57 am

I pretty much just relived that spacetime discussion, i.e. I just reread it.

I still think any disagreement boiled down to semantics. I created a page on geometrodynamics that I hope we can fill with some content (after vetting away from the nLab, i.e. the nLab is for polished material).

35. quantummoxie Says:
April 27, 2010 at 6:23 am

We could do it here if you want. I could give you posting access and we could start a page on it.

Incidentally, there is an interesting article on gravity over at FQXi. It’s about loop quantum gravity, but I seem to recall that geometrodynamics is a historical antecedent of LQG. I don’t know enough about LQG to say for certain, though.

36. Hi Steve, glad to meet you here!

Hi Ian,

You are right, geometrodynamics is related to LQG. More precisely, as part of the geometrodynamics program, Arnowitt, Deser and Misner provided a Hamiltonian for General Relativity. That is, an explicit time evolution in General Relativity. To canonically quantize a “classical” theory, you need a Hamiltonian formulation. This Hamiltonian was used in Wheeler’s and deWitt’s Quantum Gravity (quantum geometrodynamics).

The ADM Hamiltonian had some issues, e.g. its constraints were not closed under Poisson brackets, and the equations had a complicated dependence in the basic variables. Abhay Ashtekar provided new variables for the Hamiltonian. Consequently, the equations become polynomial, and the Poisson brackets closed. Moreover, his formulation, being more like Yang-Mills’s theory, allowed a direct quantization in terms of Wilson loops (Rovelli and Smolin).

In Ashtekar’s formulation, the gauge group is SO(3), or SU(2), depending on the formulation and of whether you include fermions or not. One of the variables is a connection, and the other is analogous to the “electric” field conjugate to that connection.

Wilson loops are in fact the traces of the holonomy of a gauge connection, and are gauge invariants. They work fine with all gauge theories.

• Steve Dufourny Says:
June 1, 2010 at 4:54 am

Thanks dear Christi.
It’s a beautiful and relevant blog.

Best Regards to all.

37. quantummoxie Says:
April 29, 2010 at 8:24 pm

Thanks for the overview, Cristi. I’m actually reading some of Rovelli’s stuff right now, though it’s not on LQG…

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