What do points on the manifold correspond to in reality?

[wacki]

In SR the points in Minkowski space correspond to events. I recently read in a GR lecture note that the points on the manifold do NOT correspond to events like in SR (the author even says the points don't have a direct physical meaning). So what do they represent then? And if I continuously change the Minkowski metric to whatever metric, where do I loose the correspondence between points of the manifold and events?

 

[FroChro]

I think the statement is wrong. Maybe the author meant it in some subtle way. Anyway, who is the author and is it possible to read the notes somewhere on the web?

 

[atyy]

Even in SR the points on the "blank" manifold do not correspond to events.

http://arxiv.org/abs/0802.4345
"From a General-Relativistic point of view, Minkowski space just models an empty spacetime, that is, a spacetime devoid of any material content. It is worth keeping in mind, that this was not Minkowski’s view. Close to the beginning of Raum und Zeit he stated: In order to not leave a yawning void, we wish to imagine that at every place and at every time something perceivable exists."

In GR, it is difficult (impossible?) to construct there are local gauge-invariant observables unless matter is present.

http://arxiv.org/abs/gr-qc/0110003
"If there is matter we can localize things with respect to the matter. For instance, we can consider GR interacting with four scalar matter fields. Assume that the configuration of the fields is sufficiently nondegenerate."

http://arxiv.org/abs/gr-qc/9404053
"We will refer to such gauge invariants as observables. ... Of course, these observables are not of the type with which we are familiar from, say, scalar field theory and they may be highly non-local. ... If the density ω is distributional, the resulting gauge invariant may be effectively local on M. A simple example of such an observable is the value of some scalar quantity at a point specified by an “intrinsic coordinate system.” For gravity coupled to a set of scalar fields"

http://arxiv.org/abs/gr-qc/9509026
"The use of material reference systems in general relativity has a long and noble history. Beginning with the systems of rods and clocks conceived by Einstein [1] and Hilbert [2], material systems have been used as a physical means of specifying events in spacetime and for addressing conceptual questions in classical gravity. ... The original systems of rigid rods and massless clocks discussed by Einstein and Hilbert represent unphysical idealizations. Since their time, attempts have been made to remedy this shortcoming by developing a more physically realistic description of the reference medium."

 

[FroChro]

Points on the blank manifold *do* represent events. Yes, from the point of view of general relativity, Minkowski spacetime is solution to an empty space, which is physically uninteresting situation, but it is an good and very useful approximation.

Moreover STR and GR are just theories, they do not have to be prefect. Lastly, your objection is totally void unless you define what an event and space is more precisely.
I agree there are many things to discuss about these two concepts, but in GR the points on manifold are actually postulated to represent points of spacetime which in turn do represent events I believe.

And OP was talking about GR. In GR you can have matter fields or particles that are consistent with geometry and which are lying on the manifold. So what is the problem with events then?

 

[wacki]

Thanks guys for your replies!

I can give you the link to the notes

http://www.physik.uni-wuerzburg.de/~hinrichsen/teaching/Skripte/art.pdf

but unfortunately they are in german. (page 127 for people who bother to read)
He devotes a whole paragraph to the issue. I'll give a translation of his main argument:

In SR a solution to the equations of motion can be transformed to another solution by Lorentz transformation (active transformation not coordinate change). In GR it's the same just that the transformation group is bigger (all diffeomorphisms of the manifold onto itself). The problem is that you can construct diffeomorphisms that leave one part of the manifold unchanged and map another part in a non trivial way. Imagine you have a space like hyper surface splitting the manifold in two parts. The initial conditions of a trajectory may lie in the part that is unchanged by the diffeomorphism. Let the trajectory end in the part of the manifold that is mapped non trivially by the diffeomorphism.
How can you get with a deterministic theory two different trajectories with the same initial conditions? The solutions look different on the manifold, but they actually describe the same physical situation.
Conclusion: (in bold letters) The points on the manifold have no direct physical meaning. You can not identify the points of the manifold with events.

This seems quite fundamental/basic, but it was new to me (I'm learning GR just for fun). I haven't read/noticed it in other books. However the argument is not a concrete example and a bit abstract (for my flavour). The explanation above leaves me a bit uncomfortable and I was hoping someone in this forum would have a more intuitive explanation.
(or explains why the statement is wrong).

atyy thanks for the links, it'll take me a while to go through it.

 

[Dale]

I recently read in a GR lecture note that the points on the manifold do NOT correspond to events like in SR (the author even says the points don't have a direct physical meaning).

They do correspond to events. Perhaps the lecture notes were referring to coordinates, not points.

 

[julian]

It's called "Einstein's hole argument" and not near enough people are aware of it. I'm writing a book on GR where the first chapter is on this. Einstein and Hilbert both had their own versions of it...Hilbert's is less abstract and goes like this...

Einstein demanded that the laws of physics should take the same form in ALL reference systems, that is we have the SAME differential equation to solve in all reference systems. Say you have two systems, one described by x-coordinares, the other by y-coordinates...as soon as you find a metric function g_{ab} (x) that solves the EQM in the x-system SIMPLY write down the same function but replace x with a y! This solves the differential equation in the y system! Now for these two solutions the metrics have the same functional form but they belong to two different coordinate systems so they impose different spacetime geometries! Here comes the problem: what if the two systems coincide at first but at some time after t=0 you allow them to differ? You then have two solutions, they both have the same initial conditions yet they impose different spacetime geometries after t=0! Einstein recoiled at this and spent a few years trying to get out of it but failed and eventually returned to it and resolved it before Hilbert...physicists rule!

 

[julian]

Cont...so if you accept the hole argument the you have to accept that GR DOES NOT determine the distance between spacetime points! This is how they loose their objective physical meaning...how do you refer to points? You have an origin and define other points using distance with respect to it...you can't do that anymore.

So what was Einstein's resolution? Well first you realize that if the two metrics have the same functional form then they assume all the same values they just assume them at different points - these physically equivalent solutions are related to each other by dragging the original field over the spacetime manifold! This what a diffemorphism ACTUALLY IS (NOT a coordinate transformation as most people seem to think)... Now what if you have a matter field as well...what IS preserved under a diffeomorphism are the coincidices between the values of the grav fireld and the matter field cus they are simultaneously dragged together! From this you can form a notion of the matter field being localised with respect to the gravitational field...

...this is the whole point of having PHYSICAL material reference systems, they get dragged too, and GR predicts the realtionships between the numbers registered by these material reference systems and the state of the system under study.

 

[julian]

Yes a solution of Einstein's equations gives a geometry but the solution is not unique and in actual fact there is an infinite equivalence class of geometrically distinct solutions that wipe away any objective meaning to abstarct points of the spacetime manifold. Special realivity taught us that spacetimes points have meaning up to inertial frames, in GR position and motion have become completely relative.

Einstein's version of the hole argument basically says GR gives no prescription for how grav and matter fields are localised over the spacetime manifold...All GR can do is say how physical fields are locaised with respect to each other.

You can find this stuff explained in Rovelli "Quantum Gravity" or pg 311 of "physics meets philosophy at the Planck scale".

It's a simple argument and should be in any book on classical GR/lecture course but isn't. I did PhD in condensed matter physics and none of the lecturers at this top uni had heard of Einstein's hole argument. It's been around for 95 years...you would have though people would have clocked on by now...did I just say that?

 

[julian]

Going off topic a bit but here is the great thing about this aspect of GR - cus small and large distances are gauge equivalent quantum General Realtivity, done properly, should be manifestly finite.

 

[julian]

The relation to SR?! Well as you might imagine the hole argument with its infinite equivalence class of solutions is intimately related to the question of observables in GR. Up to recently only a handfull of observable existed...in asymptotically flat spacetimes diffeomorphisms always reduce to active Poincare transformations at infinity, and hence the 9 Poincare charges at infinity are observables. BUT recently Dittrich (based on ideas of Rovelli) has provided the first systematic perturbative scheme for calculating observables of GR. In particular she has shown how GR, with its radical revision of physics, reduces to familiar physics over a fixed background spacetime when the dynamics of gravity can be ignored.

 

[julian]

I can give you the link to the notes

http://www.physik.uni-wuerzburg.de/~hinrichsen/teaching/Skripte/art.pdf

In SR a solution to the equations of motion can be transformed to another solution by Lorentz transformation (active transformation not coordinate change).

In my book I try to draw analogies with SR. I define an active Lorentz transformation as when you take a solution of say Maxwell's equations in the x-inertial frame and replace all the x's by y's. This too solves Maxwell's but is obviously physically distinct from the original solution.

I call ordinary Lorentz transformations passive Lorentz transformations - these are the transformations we know and love which connect the same physical situation in different inertial frames using the principle of the constancy of the speed of light.

Coordinate transformations are simply a relabelling of points which is purely mathematical and whose transformations are not derived from some physical principle.

 

[marcus]

Your approach is really engaging. The idea of starting with the hole argument, looking with fresh eyes, asking questions in a fresh way (not fogged over by conventional unstated assumptions.) I hope you can keep up the spritely surprise-filled style as you drive further into the subject-matter.

 

[wacki]

Thanks a lot Julian!
That was really helpfull. I can’t believe that such a fundamental point is left out in standard textbooks. I learn with Carroll and Misner and couldn't find it in those. So when is you book ready :-)

 

[julian]

Thank marcus.

I'm going off point again, but here's something else interesting coming from this stuff: GR teaches us that we MUST use real material reference systems - rods and clocks, but as they are real they will be subject to quantum fluctuations as well as the system under study. This will lead to the introduction of new fundamental decoherence into nature. Pullin and Gambini have argued that this new decoherence may resolve the measuremet problem of quantum mechanics without the shortcommings of the usual decoherence argument. It seems that nature may be imposing its own resolution of the measurement problem! Pulin Gambi et al have even formulated an interpretation of quantum mechanics based on this idea: the "Montevideo interpretation".

 

[julian]

Thanks Wacki.

The contents of my book are:

chap 1. Einstein's hole argument. Complete, Partial and Dirac observables.
chap 2. Introduction to GR and its physical observables.
chap 3. Black holes - Event, Isolated and Dynamical Horizons.
chap 4. Cosmology.
chap 5. Hawking Penrose Singularity theorems.
chap 6. Consistent Discrete Classical GR.
chap 7. Quantum field theory on Curved Spacetimes.
chap 8. Introduction to Quantum Gravity.

I keep moving from one topic to another and so no part is complete yet. At the moment I'm making progress with the singularity theorems. I'll let you know when some chap is in a better shape. In the book I try to marry simple arguments with filling in all the details of calculations...as you might imagine its quite long already - I'm trying to write the kind of book I would like to have read when I first started getting into this stuff.

In Carroll on page 435 he does mention, correctly, "...the theory is free of "prior geometry"..." and "This states of affair forces us to be very careful; it is possible that two purportrdly distinct configurations (of matter and metric) in GR are "the same," related by a diffeomorphism." It's slot away in an appendix. Plus he sounds a bit patronising of people who talk about diffeomorphism invariance seriously and I don't know if it's actually not him who is a bit confused.

 

[julian]

"...the theory is free of "prior geometry..." is the same as saying the theory provides no prescription for how the fields are localised over the spacetime manifold.

 

[wacki]

Thanks Julian, for the reference in Carroll! It seems to address the issue, but I’m still disappointed that it’s stuck in an appendix and mentioned in a way a beginner has almost no chance to appreciate its importance (otherwise the book is very good though).
Below are two web links about “Einsteins hole argument”, for people who are not familiar with it.

http://en.wikipedia.org/wiki/Hole_argument

http://plato.stanford.edu/archives/spr2004/entries/spacetime-holearg/

 

[atyy]

In addition to the refernces in post #3, Giulini's Some remarks on the notions of general covariance and background independence may be useful.

In GR, one should distinguish between "manifold" and "spacetime". Spacetime is a manifold with a metric. Both SR and GR have spacetimes and are "generally covariant". The difference is that GR has "no prior geometry": spacetime curvature in GR depends on the distribution of matter, and the distribution of matter depends on spacetime curvature.

An event in SR and in GR requires matter for its definition, since it is the intersection of worldlines. In SR you can put matter anywhere you want without changing spacetime, but in GR you cannot. In GR, it is difficult (impossible?) to construct local observables without matter, since even geometrical scalars like R(x) depend on x, but "x" has "no meaning". The convention of using test particles in vacuum spacetimes in GR is an approximation which assumes the test particles do not significantly contribute to spacetime curvature.

 

[julian]

I think Einstein himself stressed that events are physically identified by matter in SR and GR.

Yes I think in the absense of matter there are no local observables...I mean you can try asking how some degrees of freedom of the grav field correlates to the other grav degrees of freedom but the observables generated can't be 'point-like' and have to be associated with higher dimensional 'surfaces'...this is obvious as a subset of the grav degrees of freedon can't correspond to a complete description of 4d geometry.

Yep it is important to distinguish the manifold from spacetime...when I say spacetime manifold I'm referring to the manifold (blank canvas) upon which spacetime is built by imposing a metric.

 

[julian]

I vaguely remember papers in which people actually try and use degrees of freedom of the grav field as their reference systems.

 

[julian]

Hi Wacki

In Misner they don't mention the hole argument but...

As I've explained what GR predicts are the relationships that exist between quantities that can be measured...

In Misner they do the ADM or Hamiltonian formulism. From the form of the 'Hamiltonian' you discover that GR is a fully constrained theory...(thus setting it aside from normal physical theories) and it is by solving the constraints that you find out how measurable quantities are related to each other - making it a relational type theory.

 

[wacki]

Thanks Julian, for all your replies!
Let me know when your book is ready.

There is another thread in this forum related to the hole argument and diffeomorphisms
called "Active diffeomorphism". However I found your explanation a bit clearer.

And below is a short doc about the hole argument in the concrete example of the Schwartzschild metric
http://faculty.luther.edu/~macdonal/HoleArgument.pdf
This might be insightful for people who don't live and breath GR.

 

[atyy]

I vaguely remember papers in which people actually try and use degrees of freedom of the grav field as their reference systems.

I thought so too, and after reading your comment I decided to try to find where I had read it. The Rovelli reference in post #3 says it is possible but impractical, and references: PG Bergmann, Phys Rev 112 (1958) 287; Observables in General Covariant Theories, Rev Mod Phys 33 (1961) 510.

 

[julian]

I thought so too, and after reading your comment I decided to try to find where I had read it. The Rovelli reference in post #3 says it is possible but impractical, and references: PG Bergmann, Phys Rev 112 (1958) 287; Observables in General Covariant Theories, Rev Mod Phys 33 (1961) 510.

Yeah, it's my recollection that they considered it impractical, though I couldn't remember the reference. Thanks.