What is the concept of no prior geometry in GR?

[bhobba]

In a thread that is now closed in the QM sub-forum someone wrote:

Isn't spacetime supposed to be fixed, "it just is", not changing or evolving?

To which I replied:

These days we know the core of GR - its simply this - no prior geometry (it's dynamical) which is the exact opposite of the above.

But Peter Donis, who I know knows GR well posted:

No, it isn't. "No prior geometry" means the geometry depends on the distribution of stress-energy, via the Einstein Field Equation; there is no fixed geometry built into the laws of physics, the way there is in SR. It doesn't mean that the geometry "changes"; any given solution of the Einstein Field Equation describes a geometry, period; it doesn't describe one geometry "changing" into another.

I gave his answer my like because its an opportunity to fix a possible misconception I have, but for me what I said is what I thought no prior geometry meant and that geometry itself is dynamical - ie has its own Lagrangian.

I pulled out my copy of Wald and went to appendix E page 453 where the simplest possible Lagrangian embodying that, Lg = √-g R is given. Of course the full Lagrangian contains matter and fields with Lagrangian Lm so you get the total Lagrangian

L = Lm + √-g R

R is the curvature scalar and is the simplest scalar you can write embodying the curvature - its very elegant for it to be the Lagrangian. I don't know what to call it other than no-prior geometry - its seems a common name for the idea from a number of texts eg MTW uses it:
https://physics.stackexchange.com/q...o-prior-geometry-father-50-years-of-confusion

We know Einstein was wrong when he used General Covarience as the basis as pointed out by Kretschmann. Instead it became the principle of general invariance - something slightly different - but still that there is no set background geometry is the key.

You take the variation and you end up with the EFE's - the variation of Lm is defined as the stress energy tensor.

At one time I was heavily into GR - but it was a while ago. Where have I gone wrong? Was it my wording? What would be better wording if that's the case?

Thanks
Bill

 

[FactChecker]

I'm certainly no expert, but I imagine GR space-time as something that has to be determined on its own using the current metrics, curvatures, etc.. No reference to any prior geometries helps. In fact, I imagine it would be very difficult to describe the current GR space-time in terms of any prior geometry.

 

[PeterDonis]

geometry itself is dynamical - ie has its own Lagrangian

This is correct. But in the context of the other thread, it seemed like you were claiming that the geometry "changes", which seems like a reasonable inference from the word "dynamical". But in classical GR, the geometry doesn't "change"--the "dynamics" of the geometry is not telling you how the geometry "changes". It's just telling you how the geometry is determined from the field equation (as opposed to being fixed by some prior law of physics). But given a particular solution of the field equation, the geometry produced by that solution is just one geometry--it's a single 4-dimensional geometric object. So "dynamical" is probably an unfortunate choice of words in this context, but that's the word that is often used in the physics literature, so we just have to deal.

 

[bhobba]

But in classical GR, the geometry doesn't "change"--the "dynamics" of the geometry is not telling you how the geometry "changes". It's just telling you how the geometry is determined from the field equation (as opposed to being fixed by some prior law of physics).

Got it

This whole area is full of that sort of thing - wording that while common is unfortunate, like observer in QM.

One book on GR that goes big time into examining that sort of thing is the following:
https://www.amazon.com/dp/0393965015/?tag=pfamazon01-20

It takes an entirely different approach, getting linearised GR in flat space-time, then deriving the full equations from the principle of invarience (not covarience which the author is highly critical of), implying curved space-time. It was the first book on GR I read, then I read Wald. Two entirely different approaches - one downplaying geometry, while Wald embraces it. However I have to say as far as QFT is concerned Ohanian's approach is the more natural and how I explain to anyone that asks how does one go from a QFT theory of gravity to curved space-time. Of course you can only have an effective QFT of gravity - but how curved space-time comes about we know at least at a mathematical level. But the geometrical approach is just so damn beautiful and elegant. As a person with a math background Wald was elegant and beautiful - Ohanian kludgy and somewhat ugly.

What it means - I don't know - it may be telling us something very deep or simply a mathematical quirk. When I posted on sci.physics.realativity Steve Carlip explained it as its impossible to tell the difference between a theory where things act as if space-time was curved which is what Ohanian basically is and space-time actually being curved.

Thanks
Bill

 

[Paul Colby]

Sorry, I don't get this[1]. Saying there is only one geometry and it doesn't change is like saying there is only one EM field and it doesn't change.

[1] Not surprising or one of the bigger tragedies.

 

[PeterDonis]

Saying there is only one geometry and it doesn't change is like saying there is only one EM field and it doesn't change.

Yes, and both statements are true, if you understand the context in which I made them. "There is only one EM field" is correct as long as you understand that by "EM field" we mean "the full 4-dimensional solution for the EM field", i.e., an assignment of an EM field to every single event in a 4-dimensional spacetime. A full solution of Maxwell's Equations gives you that, so a full solution of Maxwell's Equations gives you "one" EM field, that doesn't "change" in that spacetime--its value at any given event in the spacetime is fixed. "There is only one geometry" has the same meaning: the geometry at any given event in the spacetime is fixed.

 

[Paul Colby]

the geometry at any given event in the spacetime is fixed.

Well, this seems a pretty vacuous statement in itself to me. When mapped out in time everything is fixed. How could it fail to be otherwise? Geometry in GR is described by the curvature which for any actual observer changes with time at some level. Yes, one can be idealized observers and universes where static geometry is possible, but not in ours.

 

[PeterDonis]

When mapped out in time everything is fixed.

Exactly--and that is what a "spacetime" is. It is "mapped out in time".

Geometry in GR is described by the curvature which for any actual observer changes with time at some level.

The curvature an observer measures in his immediate vicinity can change with proper time along his worldline, yes. But that's not what we're talking about here.

one can be idealized observers and universes where static geometry is possible

That's irrelevant to what I'm saying. "Fixed" here means "we have some particular solution of the Einstein Field Equation". That's all it means. It doesn't mean the solution is static--the solution doesn't have to have a timelike Killing vector field. It just means that we have one particular solution, and that solution describes one 4-dimensional geometry. It doesn't describe one geometry "changing" into another. It just describes one geometry.

 

[Paul Colby]

It just means that we have one particular solution, and that solution describes one 4-dimensional geometry.

Okay, that seems straight forward.

So is it in poor taste to view this global 4-geometry as an evolving local geometry we inhabit? I'm probably not bright enough to address the causality can or worms, but I've always viewed our local geometry as due to EFE as applied to the distribution and dynamics of the stress energy in our causal past, not from that in our future.

 

[PeterDonis]

is it in poor taste to view this global 4-geometry as an evolving local geometry we inhabit?

It's not a matter of "taste". I've already agreed with you that, from the viewpoint of any particular observer, moving along his particular worldline in spacetime, the geometry in his local vicinity will be "evolving", in the sense that it will be changing with respect to his proper time (as long as the spacetime isn't static and his worldline isn't an integral curve of the timelike Killing vector field). But that's irrelevant to what I was talking about when I made the statement that you originally objected to.

 

[PeterDonis]

I've always viewed our local geometry as due to EFE as applied to the distribution and dynamics of the stress energy in our causal past, not from that in our future.

You can always view the geometry at a particular event as "determined" in this way, yes. But that's also irrelevant to what I was talking about when I made the statement that you originally objected to. You're talking about causal relationships; but I (and the OP of this thread, which I was responding to) were talking about the use of the word "dynamical" in contrast to the term "prior geometry", which has nothing to do with causality; it's just ways of describing the overall structure of the theory in ordinary language.

 

[stevendaryl]

This is correct. But in the context of the other thread, it seemed like you were claiming that the geometry "changes", which seems like a reasonable inference from the word "dynamical". But in classical GR, the geometry doesn't "change"--the "dynamics" of the geometry is not telling you how the geometry "changes". It's just telling you how the geometry is determined from the field equation (as opposed to being fixed by some prior law of physics). But given a particular solution of the field equation, the geometry produced by that solution is just one geometry--it's a single 4-dimensional geometric object. So "dynamical" is probably an unfortunate choice of words in this context, but that's the word that is often used in the physics literature, so we just have to deal.

Well, from the point of view of 4-dimensional spacetime, nothing is dynamical, so geometry is not in a different boat than, say, the electromagnetic field.

[Edit]: Oh, I see that that point has already been made. Nevermind, then.

 

[bahamagreen]

In a thread that is now closed in the QM sub-forum someone wrote:

Isn't spacetime supposed to be fixed, "it just is", not changing or evolving?

Bill

That someone was me, asking from an intuitive layman's perspective. I was trying to confirm the "no change" idea of spacetime, because I was wanting to ask some more confirming questions:

- Is the "one classical outcome" (measurement problem and HUP) required* by spacetime?
- Does spacetime have no absolute orientation direction with which to define causality?

* or defined by, implied by, entailed by, consistent with, expected of... not sure the best word for this relationship

 

[PeterDonis]

Is the "one classical outcome" (measurement problem and HUP) required* by spacetime?

No, because quantum field theory can be formulated with spacetime as a background (usually it's Minkowski spacetime, but any solution of the EFE will do), and QFT is compatible with all interpretations of QM, just like non-relativistic QM is.

Does spacetime have no absolute orientation direction with which to define causality?

This is best approached in several stages:

(1) First we need to be clear about what "orientation" we are talking about. At each event in spacetime, there is a "light cone"--the set of null geodesics passing through that event. The light cone plus its interior thus contains all the events that are null or timelike separated from the given event--i.e., all the events that can possibly be causally connected to the given event. The light cone has two distinct "halves" (just imagine a double cone with the apex at a point). Considering just that event alone, the choice of which half of the light cone to label "future" and which half to label "past" is arbitrary; the math doesn't force a choice on you.

(2) Second, we need to consider what happens when we look at how the light cones "match up" from event to event. In flat Minkowski spacetime, all of the light cones "point" in exactly the same direction (I'm being heuristic here, hopefully you understand what I mean). So once we make a choice of which half of the light cone is "future" at one event, there is an obvious way to extend that choice over all of the spacetime. It turns out that there is a wide class of spacetimes (called "time orientable") in which this can be done--i.e., even though the spacetime is curved, so the light cones at different events might be "tilted" with respect to one another, the "tilt" never gets so bad that you can't make a choice of the "future" half of the light cone at one event and not have it extend in the obvious way through all of the spacetime. AFAIK every spacetime that is considered physically reasonable in GR is time orientable. But the choice of time orientation is still arbitrary, as in #1 above.

(3) Third, we need to consider how the mathematical model of a time orientable spacetime matches up with our physical observations. Obviously, we want the labeling of the light cones in the model to match up with our intuitive sense of causality; we do that by picking an event, and labeling the light cones at that event so the "past" cone contains events that we know causally influence the chosen event, and the "future" cone contains events that we know the chosen event can causally influence. In other words, since the math gives us freedom to make the choice either way, we just make it in the way that obviously works.

I'll leave it to you to decide whether all of that amounts to a "yes" or "no" answer to the question you asked.

 

[vanhees71]

No, because quantum field theory can be formulated with spacetime as a background (usually it's Minkowski spacetime, but any solution of the EFE will do), and QFT is compatible with all interpretations of QM, just like non-relativistic QM is.

Well, this is a bit overoptimistic. Usually, it's not so easy to make sense of a QFT in a general given GR background spacetime. Already an apparently simple case as the Schwarzschild solution makes trouble.

The reason is that in relativistic QFT you want to make sense of the formalism in terms of particles, which are defined as (asymptotic free) Fock states, and only in the most simple background spacetimes (e.g., de Sitter or anti-de Sitter) this is unproblematic. The classical review for this fascinating issue is

B.S. DeWitt, Quantum Field Theory in Curved Spacetime, Phys. Rept. 19, 295 (1975)
https://doi.org/10.1016/0370-1573(75)90051-4

A good textbook is

S.A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time, Cambridge University Press (1989)

 

[PeterDonis]

this is a bit overoptimistic

Yes, but I don't think the complications you mention affect the question @bahamagreen was asking. Unless we end up with a theory of quantum gravity that gives us a way to rule out some interpretations of QM.

 

[vanhees71]

True, but note that QFT of particles in a curved "background spacetime" is much less complicated a problem than to get a complete theory of quantum gravity, and this is a real physics problem (imho still the most challenging problem of contemporary theoretical physics) and not merely a philosophical quibble about interpretational problems. Imho the socalled "interpretational problems" of contemporary QT are pseudoproblems, while the real problem is indeed to find a consistent quantum theory including the gravitational interaction, and it may well be that its solution enforces another "revolution" leaving our today's theories as mere approximations to a more comprehensive new theory.

 

[bhobba]

Imho the socalled "interpretational problems" of contemporary QT are pseudoproblems, while the real problem is indeed to find a consistent quantum theory including the gravitational interaction, and it may well be that its solution enforces another "revolution" leaving our today's theories as mere approximations to a more comprehensive new theory.

Exactly - that's is the real issue. We need to lift the veil beyond the Plank scale and understand gravity totally. QM issues are IMHO are just philosophical quibbles. I find them interesting to study but think the Ensemble interpretation is perfectly fine.

Of course its just a matter of opinion - if these foundational issues in QM float your boat then like Bell you may make a breakthrough - you may even solve the gravity problem as well - although I believe the reverse is more likely.

Thanks
Bill

 

[vanhees71]

For me Bell's work on the QT foundations is almost miracolous ingenious to have found exactly the right balance between what philosophers find interesting about QT and to translate their quibbles about the ontological and science-theoretical implications (soft humanities) to a truly scientific question that is subject to empirical tests (hard science). For me the main merit of this part of Bell's scientific work is that he brought unclear philosophical concepts which also haunted many of the best physicists of their time (including Einstein and Schrödinger) into a scientifically testable prediction and thus made it a reputable science topic for physicists. Before his work it was almost career destroying, if a young physicist got involved into these foundational questions. This was to some degree justified since there was no firm scientific basis to scientifically, i.e., empirically decide between the predictions of quantum theory and inseparability due to the possibility of quantum entanglement. On the other hand to forbid people to think about a topic, and be it quite speculative at the first glance, is never a good practice in science since progress can be made only when also unconventional thinking has sufficient freedom to prosper. In this sense Bell's work on the foundations was a liberation for quantum foundational research and it had a lot of fruitful results, including the development of a new branch of physics (quantum informatics, quantum cryptography), and in my opinion it also has put to rest almost all of the philosophers' quibbles showing that nature indeed behaves as "weird" (which just means in a way unusual for our "common sense" developed from every-day experience with macroscopic objects which effectively and FAPP behave almost always classically) as QT predicts, particularly that Nature behaves probabilistic and at the same time admits stronger long-range correlations than any classical local theory allows, being itself completely based on local interactions and obeying the linked-cluster principle. At least the latter is a prerequisite for scientists to have a chance to figure out the laws of nature with well-separated systems simple enough to use them for observations and theoretical description.

 

[Paul Colby]

Well, from the point of view of 4-dimensional spacetime, nothing is dynamical

Yes, agreed. However it's the appearance of the word "dynamical" in this discussion which I found confusing reading this thread. Dynamical has a technical meaning which is close to the meaning an innocent bystander might give it. By not dynamical one is saying there is no 5'th dimension in which space-time evolves.

 

[PeterDonis]

from the point of view of 4-dimensional spacetime, nothing is dynamical

In the ordinary sense of that word, yes. But "dynamical" is also used in the GR literature to refer to the fact that the 4-d geometry of spacetime, which is not "dynamical" in the ordinary sense, is not built a priori into the laws of physics; it's determined by the particular solution of the laws that is realized. So, as I pointed out in post #3, the word "dynamical" is not really a good choice in this particular context, but we can't go back and change all the literature to pick a different one now.

it's the appearance of the word "dynamical" in this discussion which I found confusing reading this thread

Yes, that's why I pointed out in post #3 that "dynamical" is an unfortunate choice of words in this particular context. See above.