Subject: International Society for the Advanced Study of Spacetime
Date: Fri, 19 Mar 2004 21:26:19 +0200
From: Dimi Chakalov <>
To: Steven Savitt <>,

Dear Professor Savitt,

I learned about the idea of creating an International Society for the Advanced Study of Spacetime from the web site of your forthcoming conference "International Conference on the Ontology of Spacetime" (May 11-14, 2004, Concordia University, Montreal, Quebec, Canada),

I wonder if you and your colleagues would discuss the peculiar status of 3-D space in General Relativity. Please correct me if I'm wrong.

There should be no such thing as '3-D space in GR'. It should "disappear" along with time, due to the full reparametrization invariance of general relativity, as stressed by R. Parentani in gr-qc/9710130,

It seems to me that '3-D space in GR' is a Perennial, and therefore should not qualify as 'observable', according to K. Kuchar in gr-qc/9304012,

The meaning of the statement 'there is no time in GR' means that there is no preferred frame, after J. Butterfield and C. Isham, gr-qc/9903072,

Hence I'm puzzled by the question, what would be the meaning of the statement 'there is no 3-D space in GR'.

The intrinsic property of 3-D space is that we can draw a sphere around a point 'now', such that there will be a set of points inside the surface of the sphere, and another set of points outside it. Let's call the 'inside' set S_i, and the 'outside' set S_o.

This is what the 3-D space stands for. We have 'inside' and 'outside' with respect to the surface of the sphere with center the instant 'now' and radius r > 0 .

NB: However, once we lose the time (please see above), we should lose the 3-D space as well: the *individuality* of points from the two sets, S_i and S_o, must be lost. Just as we lose the individuality of spacetime points due to general covariance; they obtain their meaning only and exclusively only from the concrete physical content in some concrete and local -- but never preferred -- reference frame. Due to Diff(M) action, no physical meaning could be attached to a *bare* point on the spacetime manifold.

Paradoxically enough, we lose time but the *individuality* of points from the two sets, S_i and S_o, remains intact, just like a Perennial.

Regardless of Diff(M)-invariance which should robe their individuality, the points from S_i and S_o are *always* fixed as '3-D space' with their intrinsic properties 'inside' and 'outside'. They never conflate not exchange their fixed position, by exchanging the points of S_i with those from S_o.

Therefore, '3-D space' is a Perennial entity, and should not be
physically observable. The fact that we attach some *physical* meaning to '3-D space' is in sharp contradiction with the rules of GR. Hence we need to address the issue of the *emergence* of both time and space in quantum gravity, after C. Isham and J. Butterfield, gr-qc/9901024,

I wonder if you or some of your colleagues would like to comment. I won't be able to attend your Conference, regrettably.

More on this and other related issues at

With kindest regards,

D. Chakalov

Note: In order to "move points around", as prescribed in the Diff(M)-invariance recipe, we have to "instruct" the radius  r  (cf. above) to approach asymptotically zero. See Matthew Frank's gr-qc/0203100; web page here. Hence we treat time and space with double standards: relativistic for time, and Newtonian for space. I don't think Hermann Minkowski would have agreed with us.

In Einstein's GR, 'space' is equivalent to 'what fills space'. The notions of 'space' and 'time' must be understood as adjectives, say, 'yellow banana'. The notion of 'zero banana' corresponds to 'empty space', which has no intrinsic color (=spacetime) whatsoever. This can be best explained with Einstein's hole argument. Alan Macdonald writes [Ref. 1]: "The hole argument uses, in an essential way, the concept of a spacetime of events without a gravitational field. For the argument (tacitly) assumes that events have a physical identity independently of a metric." And yet the radius  r  (cf. above) clearly provides a finite volume of 3-D space, which exists independently of 'what fills space'. It provides a non-dynamical, fixed '3-D space per se', some pre-existing and immutable arena for "moving points around", as prescribed by the Diff(M)-invariance rules. There is no absolute time in Einstein's GR, but there is 'absolute space per se'!

We tacitly attach a very precise physical meaning to bare points on the spacetime manifold. These bare points exist 'out there' and independently from their concrete physical 'filling', like some pre-existing fixed matrices of 'absolute space per se'. For any finite  r  (see above), corresponding to infinitely many "time parameters" constituting the infamous many-fingered time, there are two spatially distinguishable sets, Sinside and Soutside, of bare points with already (Sic!) fixed spatial relations: inside vs. outside. Thus, since we begin with a finite  r  , we are allowed to set  r  to approach infinity, as in the case of the so-called asymptotically flat spacetime, hence imposing strict spatial relations,  Sinside and Soutside, to  all  Diff(M)-invariant temporal solutions on the whole 3-D hypersurface.

This is the reason why we haven't noticed the obvious Catch-22 type contradiction in describing the "evolution of gravitational systems in time": we have supplied an absolute 3-D space with fixed spatial properties, inside/outside, that are being "laid out" ahead, like Route 66. As acknowledged by A. Ashtekar, we are doing "grave injustice to space-time covariance that underlies general relativity." I fully agree. Let me recall here a similar situation with QM, in which we use math recipes that make no sense whatsoever. We use them purely as calculation tools, without being able to comprehend them at all. Likewise, the calculation machinery in GR works very well, despite the logical expectation that if we provide a tacit background of bare points with pre-fixed spatial orientation, we should run into contradictions. We have to expect something wrong to happen, and if it doesn't happen, we should be very much puzzled by the absence of any incident, as Sherlock Holmes would have said.

What made this problem of 3-D space invisible for so many years of abusing Einstein's GR with the misleading decomposition of Einstein's equations into two entirely different entities? Perhaps because there is actually a "background", but in the global mode of spacetime. Hence we 'divide Tuesday by eleven' and get fairly good experimental and observational confirmation of our math recipes. By the same token, we acknowledge the problem of time in canonical quantum gravity, but can afford the luxury of skipping the problem of 3-D space. However, we can speculate on quantum gravity in 2+1-spacetime only, and cannot solve the initial value problem [Ref. 1]. What a mess, ladies and gentlemen!

It is really incredible that we believe in some '3-D space in GR'. It's simply not there. Hence we have to zoom on the dynamical nature of infinitesimal, which requires two modes of spacetime: local and global, just like in the case of the human brain. Its evolution along the putative universal time arrow (cosmological time arrow) has been explained in my White Paper.

But since I promised that no knowledge in brain neurophysiology will be required, let me explain the paradoxical situation of  '3-D space in GR' with the so-called gravitational waves.

In the latest book by Brian Greene, "The Fabric of the Cosmos", he stressed that these putative waves do not propagate through spacetime but within spacetime (cf. p. 419; his email address is in the CC: list above).

Now, my dog knows what is the meaning of 'propagating through spacetime'. He really loves to play Frisbee at the beach, and can easily pin down the time parameter of its trajectory, as I mentioned in the front page. The corresponding task of LIGO, however, is by no means trivial, since there is no background time parameter nor background 3-D space. These "waves" propagate within themselves, and with respect to themselves (much like in the old story of von Münchausen, who managed to lift himself and his horse by pulling himself up by his hair). I'm definitely sure that Hermann Minkowski wouldn't agree with us. The paradox is known since 1917.

On the other hand, the effect from these gravitational waves, as observed in our past light cone, cannot be refuted. We can even hear it, thanks to John Cramer who fulfilled the request of an 11-year-old boy who wanted to know what the Big Bang sounded like for a school project.

The resolution of this truly paradoxical situation is to introduce two modes of spacetime, and subsequently a dual age of the universe. Again, the "gaps" needed for the elementary step of the "expansion of spacetime" are placed in the global mode of spacetime, and are hidden to the local mode of spacetime by the so-called 'speed of light'. The gravitational-and-quantum waves "propagate" only in the global mode of spacetime, and their unique effects can be calculated for one, and no more than one, instant in the local mode of spacetime. Strictly speaking, we cannot build any trajectory comprised of their effects in the local mode of spacetime, because we would need more than one instant in which the values of all physical quantities are uniquely defined. That's impossible in the present framework, since the determination of unique values is one-at-a-time only, along the universal time arrow.

We can calculate a trajectory for our Frisbee (or anticipate it, as my dog probably does) from one instant of its "instantaneous velocity", because the effects of its gravitational-and-quantum waves are vanishing small, but once we leave the scale of tables and chairs the situation changes drastically. Moreover, these effects are only observable post factum, in our past light cone only, while the gravitational-and-quantum waves are placed in the potential future of the universal time arrow (also cosmological time arrow). All the paradoxes from quantum theory and Einstein's GR are result from our efforts to "explain" the effects of these waves exclusively in the local mode of spacetime.

Back in the past century, on August 15, 1999, I suggested a paradigm for unification of the gravitational-and-quantum waves: these "waves" do not carry any energy but are manifestation of two virtual worlds, material and tachyonic. These atemporal empty waves "propagate" in the global mode of spacetime, and their cancellation produces one fixed value of all physical systems in the local mode of spacetime, one-at-a-time. There are many imprints from this paradigm left on quantum theory, but in Einstein's GR we have deliberately cut off the "negative mass" empty wave by hand. I think this is wrong. We should keep the "positive mass" and "negative mass" components of the virtual gravitational wave. I don't agree with the textbook lore that there is no dipole gravitational radiation because we observe -- in the local mode of spacetime --  only one sign of mass, called "positive", and also because we haven't yet found the additional two gravitational wave polarizations and their time derivatives, as explained by Steve Carlip. They all are in the global/vertical mode of spacetime, not in the local/horizontal mode. People think only about the latter, but then they inevitably have to consider some "quadrupole radiation", then have to postulate, with a great deal of wishful thinking, that the metric perturbation might be sinusoidal, and that the gravitational wave amplitude might produce some observable effects, provided they 'increase the sensitivity' of detectors (see a recent paper about LIGO here).

Sadly, nobody seems to care about the fundamental proof by Hermann Weyl that the energy of these "waves" cannot be observed in principle, and nobody reads the papers by Angelo Loinger either. People simply take the position of my dog watching the Frisbee, and say: hey, there has to be some observable effect of the acceleration of the Frisbee, such that the magnitude of the gravitational wave amplitude, h , produced at a "distance  r  from the Frisbee" is proportional to the second time derivative of the quadrupole moment of the Frisbee and inversely proportional to  r , while the luminosity of the Frisbee is proportional to the square of the third time derivative of the quadrupole moment of the Frisbee. They don't care that their Frisbee is only 4 per cent of the whole stuff in the universe. They just need a couple of billion dollars, taxpayers' money, to make "gravitational wave astronomy", and that's it. How can you measure a "distance  r  from the Frisbee", and in what reference frame? Are you some ideal omnipresent observer orthogonal to both space and time? If you subscribe to the notion of 3-D space in current GR, you can hardly notice the problem with "distance  r  from the Frisbee", since you will be tacitly referring to an absolute 3-D space with fixed spatial properties, inside vs. outside, as explained above. Also, you may never notice that the bi-directional talk between matter and geometry, after J. Wheeler, cannot fit into the end product of this same "talk". It requires a global mode of spacetime, again. You are aware that Einstein's GR is a genuine non-linear theory -- the metric field not only affects, but also is affected by, the other fields (John Baez) -- but you don't notice that the intrinsic timing of this negotiation is completely invisible in the end product, called 'local mode of spacetime'.

What a mess, ladies and gentlemen! How come the dog didn't bark?

Well, I'm afraid my dog would neither be able to understand the crucial distinction between the two modes of spacetime nor the dynamical nature of the infinitesimal, but I believe it will be quite comprehensible to the theoretical physicists at GR17 in Dublin. Provided they would be interested, of course.

D. Chakalov
March 20, 2004
Last update: January 6, 2005
"Is there any other point to which you would wish to draw my attention?"
"To the curious incident of the dog in the night-time."
"The dog did nothing in the night-time." 
"That was the curious incident."

Sherlock Holmes, "The Adventure of Silver Blazes"


[Ref. 1] Alan Macdonald, Einstein's Hole Argument, Am. J. Phys. 69(2), 223 (2001); pdf file here.

"2. The issues discussed here complicate the initial value problem for the
field equations of general relativity.[7, 8]

Now consider the vacuum field equations of general relativity. They are of
second order. Thus one might expect that data consisting of the metric and its first derivatives on an initial spacelike 3-manifold, with some constraints from the field equations, would uniquely determine the solution at later times. But no such data can do this: any solution can be transformed to others by coordinate transformations which leave the data fixed. The field equations cannot even uniquely determine the topology of a manifold on which a solution is defined.[9]"
[9] C. Misner, K. Thorne, J. Wheeler, Gravitation (Freeman, San Francisco, 1973), p. 837.