# Is Newtonian spacetime flat too?

## Main Question or Discussion Point

It is said that minkowksi spacetime is flat. How about galilean (newtonian) spacetime, is it flat too? If not, what is it?

It is also said that it is unknown whether there is geometry or spacetime inside planck scale. If there is none. I can't imagine how the planck scale without geometry can create the flat spacetime... although I can imagine how spin-2 field in that flat spacetime can create gravity (or curved spacetime).

According to Richard Feynman in his book "Feynman Lectures on Gravitation":

The claim that the only sensible theory of an interacting massless spin-2 field is essentially general relativity (or is well approximated by general relativity in the limit of low energy) is still often invoked today. (For example, one argues that since superstring theory contains an interacting massless spin-2 particle, it must be a theory of gravity.) In fact, Feynman was not the very first to make such a claim."

The field equation for a free massless spin-2 field was written down by Fierz and Pauli in 1939[FiPa 39]. Thereafter, the idea of treating Einstein gravity as a theory of a spin-2 field in flat space surfaced occasionally in the literature.

If Einstein gravity is a theory of a spin-2 field in flat space. How can you reconcile this with the conjecture that spacetime doesn't even exist in planck scale. What produce flat spacetime (or gallelian space) then from planck scale? Is there a corresponding field for flat spacetime? I just want to imagine how this occurs. Thanks

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haushofer
It is said that minkowksi spacetime is flat. How about galilean (newtonian) spacetime, is it flat too? If not, what is it?
No, it is not if gravity is around. Newton-Cartan theory provides a geometric reformulation of Newtonian gravity in the language of metrics, connections and Riemann tensors. The corresponding spacetime is curved in the "time-space" direction, while the spatial foliations are flat.

So, if you take a closed loop in this spacetime in the time AND space direction, you will find curvature. Mathematically this is expressed by the fact that the only non-zero component of the Riemann tensor is

$$R^i{}_{0j0} = \delta^{ik} \partial_k \partial_j \phi$$

where phi is the Newton potential.

It is also said that it is unknown whether there is geometry or spacetime inside planck scale. If there is none. I can't imagine how the planck scale without geometry can create the flat spacetime... although I can imagine how spin-2 field in that flat spacetime can create gravity (or curved spacetime).
Newton-Cartan is a reformulation of Newton's theory, which doesn't know about Planck scales (it's an effective theory, which breaks down way below this scale).

You could check Misner, Thorne and Wheeler for a review on Newton-Cartan theory, or

http://arxiv.org/abs/1011.1145

I hope this helps :)

It is said that minkowksi spacetime is flat. How about galilean (newtonian) spacetime, is it flat too? If not, what is it?
First of all, in Newtonian or Classical mechanics, Space and Time were still both separated, as Newton believed that time was invariant - a dimension that is independent and constant. So they only had basically a "Newtonian or Classical Space" instead of a 4D spacetime. Newtonian space was three-dimensional in so far as it uses the inverse-square law, as well as Gauss' theorem and divergence. The R2 in the law was derived from the surface area of a sphere (4πr2), which was - and still is - the three-dimensional model of a celestial body, such as Earth, planets, and stars. The inverse-square law only works in 3D Euclidean space.

It is also said that it is unknown whether there is geometry or spacetime inside planck scale. If there is none. I can't imagine how the planck scale without geometry can create the flat spacetime... although I can imagine how spin-2 field in that flat spacetime can create gravity (or curved spacetime). If Einstein gravity is a theory of a spin-2 field in flat space. How can you reconcile this with the conjecture that spacetime doesn't even exist in planck scale. What produce flat spacetime (or gallelian space) then from planck scale? Is there a corresponding field for flat spacetime? I just want to imagine how this occurs. Thanks
In Quantum Mechanics, They have what is called "Quantum Foam" - a concept which describes the structure of spacetime at the Planck/subatomic scale.

"Quantum foam (also referred to as spacetime foam), is a concept in quantum mechanics devised by John Wheeler in 1955. The foam is supposed to be conceptualized as the foundation of the fabric of the universe.

Additionally, quantum foam can be used as a qualitative description of subatomic spacetime turbulence at extremely small distances (on the order of the Planck length). At such small scales of time and space, the Heisenberg uncertainty principle allows energy to briefly decay into particles and antiparticles and then annihilate without violating physical conservation laws. As the scale of time and space being discussed shrinks, the energy of the virtual particles increases. According to Einstein's theory of general relativity, energy curves spacetime. This suggests that - at sufficiently small scales - the energy of these fluctuations would be large enough to cause significant departures from the smooth spacetime seen at larger scales, giving spacetime a "foamy" character."

More at:http://en.wikipedia.org/wiki/Quantum_foam

bcrowell
Staff Emeritus
Gold Member
No, it is not if gravity is around. Newton-Cartan theory provides a geometric reformulation of Newtonian gravity in the language of metrics, connections and Riemann tensors. The corresponding spacetime is curved in the "time-space" direction, while the spatial foliations are flat.
I think it would be more accurate to say that Newtonian gravity can be expressed in more than one way. In Newton's formulation, spacetime is flat. In the Newton-Cartan formalism, spacetime is curved.

We get something similar with GR, where it's possible to describe gravity as a spin-2 field on a Minkowski background, and you end up with something that most theorists believe is equivalent to GR: Deser, Gravity from self-interaction redux, 2009, http://arxiv.org/abs/0910.2975 But once you complete the theory in a self-consistent way, the original flat background is no longer detectable.

julian
Gold Member
It is said that minkowksi spacetime is flat. How about galilean (newtonian) spacetime, is it flat too? If not, what is it?

It is also said that it is unknown whether there is geometry or spacetime inside planck scale. If there is none. I can't imagine how the planck scale without geometry can create the flat spacetime... although I can imagine how spin-2 field in that flat spacetime can create gravity (or curved spacetime).

According to Richard Feynman in his book "Feynman Lectures on Gravitation":

If Einstein gravity is a theory of a spin-2 field in flat space. How can you reconcile this with the conjecture that spacetime doesn't even exist in planck scale. What produce flat spacetime (or gallelian space) then from planck scale? Is there a corresponding field for flat spacetime? I just want to imagine how this occurs. Thanks
In any gravitational field clocks run faster higher up so physically newtonian space is 'time-curved'. Infact in the Feynman lectures vol III he proves that for a weak grav, low speeds, low mass...curved time +first approx to time dialation + freely moving objects follow a path that maximises proper time + while ground clock should register the same value each time = Newtonian grav. Basically clocks can gain proper time by going higher up but they have to do so faster which slows down the clock - the comprimise corresponds to newtonian theory.

As for GR being interacting spin-2 particles...top notch GR people like Ashtekar dont believe that you can get the Schwarzschild solution, which is a kind of bound state, from interacting gravitational radiation. Hawking in "The geometric universe" pg 126 says something precise about topology change. Plus you should have a look at "From Gravitons to Gravity:Myths and Reality" by T. Padmanabhan.

As for how you get flat spacetime from a discerete quantum spacetime ...everyone doing non-perturbative gravity wants to know this....varies clever interesting ideas but wont go into them now. Could say it is the missing link in LQG.

This spin-2 thing relation to quantum gravity...it's a the very best an approximation about a fixed classical space time. Dont be fooled - this IS NOT equivalent to quantisation as it is akin to taking the classical path of a particle and quantising its fluctuations and this is not the same as Schrodinger's equation.

Perturbative quantum gravity pressuposes that spacetime can be approximated by flat spacetime...you should not assume the nature of quantum spacetime, you should let the theory TELL you what quantum spacetime is - exactly what they are doing in LQG for example.

Witten is aware of the limitations of gravitons over fixed spacetime and stated from the beginning he wanted a formulation of quantum spacetime whereby some notion of string-like excitations should actually creates the quantum space rather than being fluctuations over it.

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The metric $ds^2=-(1+2\phi)dt^2+dr^2+r^2d\Omega^2$ reproduces Newtonian gravity around a spherical mass, where $\phi=-GM/r$ is the potential.

bcrowell
Staff Emeritus
Gold Member
As for GR being interacting spin-2 particles...top notch GR people like Ashtekar dont believe that you can get the Schwarzschild solution, which is a kind of bound state, from interacting gravitational radiation. Hawking in "The geometric universe" pg 126 says something precise about topology change. Plus you should have a look at "From Gravitons to Gravity:Myths and Reality" by T. Padmanabhan.
Padmanabhan is very careful in the claims he makes. He just claims that the equivalence between the spin-2 picture and GR isn't quite as clean and unambiguous as Deser originally claimed. Deser also doesn't buy Padmanabhan's claims: http://arxiv.org/abs/0910.2975

Why would topology change be relevant? The standard formulation of GR doesn't allow topology change (or can't describe it). So if the spin-2 picture doesn't allow topology change either, how does that affect Deser's claim that the two theories are equivalent? Amazon let me see the relevant page by Hawking, and I don't see how it's relevant. (It's not even about classical gravity.)

Interesting about Ashtekar. Could you provide the reference?

Perturbative quantum gravity pressuposes that spacetime can be approximated by flat spacetime...you should not assume the nature of quantum spacetime, you should let the theory TELL you what quantum spacetime is - exactly what they are doing in LQG for example.

Witten is aware of the limitations of gravitons over fixed spacetime and stated from the beginning he wanted a formulation of quantum spacetime whereby some notion of string-like excitations should actually creates the quantum space rather than being fluctuations over it.
The OP asked a question about classical gravity. Quantum mechanics is irrelevant here.

julian
Gold Member
I see the words planck scale in his opening and how to recomcile it with flat spacetime.

Deser says he starts in Minkowski spacetime then Minkowski spacetime is undetectible...aint ever going to get the Schwarzchild solution. GR doesnt start in Flat spacetime. GR isnt claiming to bring about topolgy change...it's implied by Deser isnt it? (there is a tetrad formulation where topology chamges are allowed beside the point)

Hawking's comment is the implication of a fact of classical GR on the information paradox...the statement of topology change is purely classical.

bcrowell
Staff Emeritus
Gold Member
I see the words planck scale in his opening and how to recomcile it with flat spacetime.
Sorry, I missed that. Actually, the OP's question is completely unintelligible, but I agree that the OP does seem to think he's asking something about quantum gravity.

Deser says he starts in Minkowski spacetime then Minkowski spacetime is undetectible...aint ever going to get the Schwarzchild solution.
There's a lot missing if you want to make this into a real argument.

GR doesnt start in Flat spacetime.
I don't understand what you're claiming here.

GR isnt claiming to bring about topolgy change...it's implied by Deser isnt it?
I don't understand what your point is.

julian
Gold Member
My slight understanding was that Deser completed an infinite number of iterations started in Minkowski spacetime to get out the action of GR. So it does all start in flat spacetime doesnt it?? And the GR action is supposedly brought about by the real physical process of interacting gravitational radiation starting in flat spacetime. I know summing up a finite number of terms doesn't change Minkowski spacetime and in actual fact you have to sum up an infinite number before spacetime becomes curved...so I agree with you on that.....but I still dont like how you turn on the interaction then sudenly there is an event horizon and a topolgy change via a physical process. I'll try find that quote of Astekar. I'll have to look at Padmanabhan again. Thanks. I know that he and Deser had email exchanges but neither convinced the other. (non-standard GR and topolgy change...was that Gary Horowitz? did you read that as well?)

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julian
Gold Member
Deser assumes the physical process of interacting grav radiation starting in Minkowski brings about GR is what I meant by it starts in flat spacetime. Einstein just wrote down his equations without any considerations of this kind is what I meant by GR doesnt start in flat spacetime.