The ADM Approach: Understanding the Basics of Minkowski Space and Metric Tensors

[member 11137]

Certainly an easy question for you (specialists on this forum) but absolutely not clear for me.
The ADM Approach (1 + 3), as it can be seen in Misner Thorne and Wheeler or some other reference, is based on a kind of extension of the Pythagorras Theorem to a Minkowki space. Ok.
Its signature is (- +++); convention, I suppose; ok.
And then you can see in some equations resulting of this approach a (3-3) matrix supposed to represent the local 3D metric tensor as if it could have any value and not obligatory the unit matrix I(3) which is the spatial part of the (4-4) one associated to a Lorentz metric.

Why? Does it come from the manner to "cut" the slice of time? I am sure something is not ok in my head concerning this point. Thanks for help.

 

[HallsofIvy]

Certainly an easy question for you (specialists on this forum) but absolutely not clear for me.
The ADM Approach (1 + 3), as it can be seen in Misner Thorne and Wheeler or some other reference, is based on a kind of extension of the Pythagorras Theorem to a Minkowki space. Ok.
Its signature is (- +++); convention, I suppose; ok.
And then you can see in some equations resulting of this approach a (3-3) matrix supposed to represent the local 3D metric tensor as if it could have any value and not obligatory the unit matrix I(3) which is the spatial part of the (4-4) one associated to a Lorentz metric.

Why? Does it come from the manner to "cut" the slice of time? I am sure something is not ok in my head concerning this point. Thanks for help.

What coordinate system is being used? There is nothing "obligatory" about
I(3)! One can show that as long as the coordinate axes are orthogonal to one another the metric tensor is diagonal but even in flat space, with spherical coordinates, it is not I(3). The reason tensors were so important in general relativity is that as soon as there is mass, space is not flat and the metric tensor cannot be I(3) with any coordinate system.

 

[member 11137]

One can show that as long as the coordinate axes are orthogonal to one another the metric tensor is diagonal but even in flat space, with spherical coordinates, it is not I(3). The reason tensors were so important in general relativity is that as soon as there is mass, space is not flat and the metric tensor cannot be I(3) with any coordinate system.

Ok. But as there are masses everywhere around us here on the Earth and as there is probably a lot of energy (equivalent to mass, isn'it?) in vacuum (even if only a pair of photons per cubic kilometer), the metric tensor is never exactly of Lorentz or I(3) for the spatial part and then it seems to me incoherent or at least very difficult to built something based on the Pythagoras Theorem... I don't know if I am clear enough in my way to explain, but I am lost. I think I must re-read the demonstration of the ADM Approach. Thanks for help

 

[pervect]

I'm not, unfortunately, that familiar with the ADM approach, but the discussion in MTW seems to be reasonably clear on how the split is done into space and time.

Compare the two hypersurfaces with two ribbons of steel out of which one wishes to form a rigid structure. To give the geometry of the ribbons themselves by no means fixes that structure - for that purpose, one needs cross-connects between one ribbon and the other. It is not even enough to specify that these connections are to be welded on perpendicularly to the lower ribbon, to specify where each is welded, and to give its length. One must in addition tell where each connector joins the upper surface.

So you wind up replacing a 4-d metric with two 3-d metrics, a lapse function, and a shift function. The 3-d metrics give the metric of each hypersurface, the lapse function gives the distance between the upper and lower hypersurfaces, and the shift funciton tells us which point on the lower surface is connected to which point on the higher surface.

 

[member 11137]

I don't know if I am clear enough in my way to explain, but I am lost. I think I must re-read the demonstration of the ADM Approach. Thanks for help

So. I shall try to explain what is introducing a doubt in my head concerning this demonstration. The ADM procedure consists to define the necessary tools to connect the 3D geometry at time t in M to the 3D one at time t+dt in M + dM. There is no "a priori" concerning the first geometry nor the second one. Because of this one can expect the absence of restriction on the metric tensor (3D), specially in presence of matter (your remark). This means: the ADM construction is based on a any geometry. But why are we authorized to calculate the relation between proper time and proper length, distances, with the Lorentzian type of metric ? Thank you for the ligth.

 

[Stingray]

General relativity assumes that spacetime always posseses a metric with Lorentz signature (which is generally not Minkowski). The "Pythagorean theorem" that I think you're referring to is usually what is used to define the metric (in part). It's an assumption that is formally contained in the structure of GR.

 

[member 11137]

General relativity assumes that spacetime always posseses a metric with Lorentz signature (which is generally not Minkowski). The "Pythagorean theorem" that I think you're referring to is usually what is used to define the metric (in part). It's an assumption that is formally contained in the structure of GR.

Cartan's work tells us that any quadratic form can be reduced to a sum of squares. Your assumption is equivalent to: any quadratic form of the GR can be reduced to a sum of squares with signature - +++; correct? And so one can apply the Pythagorean theorem (thank you for the correct translation of this name and for the helps): correct? I think I begin to understand why the ADM Approach is not reducing the generality (I was afraid it could have...)

Subsidiary questions: Is there consequently no authorized transformation within the GR that permits a change in the signature? What would represent a change of the signature in the language of the geometry?

 

[Stingray]

Subsidiary questions: Is there consequently no authorized transformation within the GR that permits a change in the signature?

Correct. If the signature changed, then physics would look very different.

What do you mean about the ADM approach "reducing generality?" Their formalism (an extension of work by Lichnerowicz) derives the constraint and evolution equations of the metric (or more precisely, of the 3-metric and extrinsic curvature - i.e. the first and second fundamental forms of the hypersurfaces) from Einstein's equation. The reasons for splitting up spacetime into a bunch of spacelike hypersurfaces are to show that GR has a well-defined initial value problem and to put it into Hamiltonian form (which is useful for many things).

 

[member 11137]

Correct. If the signature changed, then physics would look very different.

What do you mean about the ADM approach "reducing generality?"

Oh my first anxiety has now disappeared, had no justification (or more exactly had justification due to a bad analyze of the situation) and was the matter of the present discussion.

But the result of our discussion is that only 3D metric tensor with a reduction -+++ are admissible within the ADM Approach. All other one are corresponding to something else not necessary related to a physical reality; or at least not to real phenomena described by the GR. This is ambarassing for my own approach that seems to embed quite more mathematical cases.

In this sense I have now the feeling that the ADM approach is reducing a mathematical generality. Or perhaps I need your help again to go further.
Thanks for the present discussion.

 

[hellfire]

I am not familiar with this, but it seams obvious to me that you are right when you say that there is a loss of generality in the mathematics of the theory, as only manifolds which can be foliated in non-intersecting spacelike surfaces can be treated with the ADM formalism. Otherwise you can have also other kind of manifolds in GR, the question is whether they are physically meaningful. Is this what you mean?

 

[member 11137]

I am not familiar with this, but ...

I am not familiar with this too (unfortunately as say Pervect).

Otherwise you can have also other kind of manifolds in GR, the question is whether they are physically meaningful. Is this what you mean?

Yes, it is what I mean; but not only. You are right when you speak about the ADM formalism (non intersecting spacelike manifolds). My way of doing (if you have followed it) represents a different approach in the sense that I am exploring as generally as possible a family of equations. The starting point is the derivation along the time of the Poynting's vector. But I left this particular case and tried to win more generality, only regarding and studying the formalism introduced by this partial derivation. 3D solutions to this mathematical question show themselves a formalism owning a beautiful analogy with some equations of the ADM approach. Thus "if the comparison is physically meaningful", I got a way to reconnect my calculations with the ADM approach. This is exactly what I am actually exploring.

It is really interesting if one consider a propagating plane EM wave as a spacelike manifold at different times in a chronology (a natural superposition of thin sandwiches). My approach seems to work correctly for the description of this kind of waves. It seems also to fit for waves that are not exactly plane. And it "seems" to fit for the ADM approach: I am still exploring this important point...

 

[Stingray]

I'm having trouble understanding your english (sorry), but have you looked at the characteristic formulation of GR? There, you specify initial data on a null hypersurface rather than a spacelike one.

 

[member 11137]

I'm having trouble understanding your english (sorry)

In fact, I am sorry for this. If you can speak French or German, please don’t be afraid to e-mail me. Our conversation will be really easier*.

... but have you looked at the characteristic formulation of GR? There, you specify initial data on a null hypersurface rather than a spacelike one.

Apparently a pretentious attitude but in reality just another motivation is leading my research. As said before, my first preoccupation was to explore the following intuition: does any cross product split in accordance with the local geometry? If yes when and how. So, at the beginning, I didn’t consider the problematic with the standard eyes of the GR. It doesn’t mean that I refuse to do it now: our conversation is a prove of that.

At each given point M and at each given time t in the universe, one can reasonably expect to encounter one and only one geometry in the proper chronology of the event (M, t) under consideration. This local topology is the common “thing” between all characteristics and properties defining what is happening on M at t; and so it is for all cross products of any type: the rot E x E, the rot H x H, the u x w and also the r x v that are angular momentum and quantized.

This preoccupation about a connection between the topology and what happens is a strategic point of view in my approach and the reason why I feel allowed to post here in “Special and General Relativity”. The connection appears under the form of a “conique (conoide)”:
A. x² + B. y² +C. z² + D. x. y + E. y. z + F. x. z + G. x. z + H. x + J. y + K. z + L = 0
I simply explore what happens if and when the spatial part of the geodesic “followed” by a photon is in coincidence with this conoide.

Sorry, I can’t explain this better* and I do not have achieved my explorations concerning the initial data problem (which is in particularly implying a time symmetric and a time anti symmetric one). As said, first motivations was centred on the exploration of the junction: (topology / splitting of a cross product) and (topology / splitting of a mathematical extension of the cross product). Certainly not very original.