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Is "Galilean gravity" a thing?


geordief
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A comment in the ongoing "Nature of Time"  thread made me wonder  what gravity would look like if the Galilean version of Relativity  was correct

 

I have seen the animations depicting Galilean spacetime diagrams  and wonder whether  or not they can be extrapolated to a model of gravity in a similar way to how AE moved on to GR from Special Relativity.

 

Just  to say,I am not attempting to attach any credence to Galilean Relativity.

I just wonder if that model might be stretched  to incorporate  a kind of Galilean General Relativity  and what "Galiliean Gravity" might look like. 

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Under GR through the Principle of General Covariance it would be represented under the Newton approximation solutions of the Einstein field equations.

The simplest transform is the Minkowskii metric, Euclidean space or flat space. This is denoted by [latex]\eta[[/latex]

 [latex]\mathbb{R}^4 [/latex] with Coordinates (t,x,y,z) or alternatively (ct,x,y,z) flat space is done in Cartesian coordinates.

 

In this metric space time is defined as

 

[latex] ds^2=-c^2dt^2+dx^2+dy^2+dz^2=\eta_{\mu\nu}dx^{\mu}dx^{\nu}[/latex]

 

[latex]\eta=\begin{pmatrix}-c^2&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}[/latex]

the metric above works well to describe The Newton limit provided provided you don't have significant relativity effects due to either inertia or extreme mass. Though these equations do describe the essentials of SR, they still readily apply for situations not involving gamma of the Lorentz transforms.

Edited by Mordred
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12 minutes ago, geordief said:

You can have gravity in flat space?(apologies for naivety or misunderstanding)

Not precisely Newton physics also includes the Newton Shell theorem. In that shell theorem a uniform mass distribution would have no gravity. However if you have an anisotropic distribution such as the the Earth where gravity is weak we don't have significant time dilation effects. so the Newton method for everyday measurements are still accurate. It is only when you get extremely fine tuned in your examination that the time dilation becomes measurable. so there is curvature aka gravity but the curvature isn't significant and we can still get good approximation under Euclidean flat geometry.

Ok lets take two falling objects you can do this with a pen and paper easily enough Draw a circle. the Center of the circle is your center of mass. Choose two angles from that center of mass say 15 degrees and 345 degrees. Let those represent the two infalling particles toward the CoM. You assign a variable to represent the separation distance between the two infalling particles at a given radius. The common symbol used is 

\[\xi\]

the value will decrease as the particles approach the CoM. this is termed tidal force due to the geometry (curved spacetime though under Euclidean approximation ) this is often described as gravity as the tidal force due to curved spacetime

 

Edited by Mordred
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3 minutes ago, Genady said:

In a homogenous isotropic universe flat space with gravity is one of the possibilities, which btw looks like what we in fact got. 

apply Newtons shell theorem a uniform mass distribution wouldn't have gravity as you wouldn't have any net force at any arbitrary coordinate treated as center of mass. Gravity itself requires non uniformity of mass distribution

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As Genady said, gravity in a Galilean universe would look like Newtonian gravity, but I don't think there would be magnetic fields. Or at least they would be different. Maxwell's equations are Lorentz invariant (that was the whole motivation for Einstein's development of relativity in the first place), so their current form wouldn't be possible in a Galilean world.

Edited by Lorentz Jr
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15 minutes ago, Genady said:

To clarify, in GR based cosmology, the flat space (together with open and closed ones) is a possible solution for a homogenous isotropic universe.

that's more accurate +1 in point of detail the flat universe geometry is extremely Newtonian albeit expanding. The metric is simply Euclidean with the scale factor representing the commoving coordinate changes (for the Flat Euclidean case)

Edited by Mordred
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In this lecture Alan Guth explains problems with applying Newton's law of gravity, including the shell theorem, to infinite space. Basically, it boils down to a conditionally convergent integral, which doesn't have a unique answer. Lecture 6: The Dynamics of Homogeneous Expansion, Part II | The Early Universe | Physics | MIT OpenCourseWare

18 minutes ago, Lorentz Jr said:

 Maxwell's equations are Lorentz invariant (that was the whole motivation for Einstein's development of relativity in the first place), so their current form wouldn't be possible in a Galilean world.

Unless, that world contains ether...

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1 hour ago, Genady said:

Unless, that world contains ether...

Right. So Lorentz invariance would be a (very strange) property of the ether (i.e. the vacuum), not a consequence of "geometry". That means it wouldn't necessarily apply to all natural phenomena, so there could be influences that propagate faster than light.

Without an ether, the Galilean limit would be where c goes to infinity, which would mean ϵ0μ0=0 . So ϵ0 and/or μ0 would have to be zero.
If ϵ0=0 , there would be no electric charge, and that means no magnetic field. So no electric or magnetic phenomena at all.
If μ0=0 , Ampère's circuital law wouldn't exist. That's the one where the magnetic field is affected by electric currents and changing electric fields,
so the magnetic field would always have to be zero (or maybe a uniform constant throughout all of space, in which case space wouldn't be isotropic, i.e. the magnetic field would define a special direction in the universe and electric charges would keep zooming around in circles and helical paths 😄).

Edited by Lorentz Jr
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23 minutes ago, Genady said:

In this lecture Alan Guth explains problems with applying Newton's law of gravity, including the shell theorem, to infinite space. Basically, it boils down to a conditionally convergent integral, which doesn't have a unique answer. Lecture 6: The Dynamics of Homogeneous Expansion, Part II | The Early Universe | Physics | MIT OpenCourseWare

Unless, that world contains ether...

ah the influence of the scale factor under commoving coordinates. Good link by the way 

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35 minutes ago, Genady said:

I think a short answer to the "gravity in flat space" question is that the gravity is a curvature of spacetime, a 4D object, rather than that of space, which is 3D.

I would have to say that's an accurate short answer

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39 minutes ago, Genady said:

I think a short answer to the "gravity in flat space" question is that the gravity is a curvature of spacetime, a 4D object, rather than that of space, which is 3D.

 So  we can talk about curved 3d space or curved 4d spacetime?

They are different models.....

I was only familiar with the concept of curved 4d spacetime.

Could you perhaps explain(or give me a pointer)  how  one would measure a curvature  in 3d space .**

I think  I understand  the principle of curvature in 4d spacetime.

 

**I suppose it must refer to the surface of a sphere?

 

 

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This is the curvature I'm talking about. It applies to any "smooth" manifold with any number of dimensions:

Riemann curvature tensor - Wikipedia

@geordief  Let me add an example of the three possibilities for the space of a homogeneous isotropic universe. Take any three points in space, far from each other, for which the distances between them can be measured. They make a triangle. Knowing the distances, calculate the sum of angles of the triangle. If the sum is 1800 then the space is flat. Otherwise, it is curved. If it is less than 1800, the space is "open" or "hyperbolic". If more, it is "closed" or "spherical". 

Edited by Genady
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Similar to the example posted by Mordred above, where the radii of test masses, in flat space, converge on the CoM ( tidal ), and we could call this convergence 'gravity', in curved space, as on a globe ( positive curvature where a triangle's angles add up to more than 180o ), two parallel lines of longitude, at the equator, converge to a point ( south or north pole ) and we can also ascribe this convergence to 'gravity'.
This can also be done with negative curvature ( saddle shape where a triangle's angles add up to less than 180o ).

In our particular case, the curvature involves space-time, which is 4D.

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 Yes that example is mathematically described a parallel transport of two vectors, which returns the scalar value for the separation distance described by

\[\xi\].  This parallel transport is equally used to define the various curvature terms. if the two parallel paths converge you have positive curvature. If they diverge then negative curvature. If they remain flat Euclidean geometry is preserved including those applicable to Pythagoras theorem. Angles as mentioned add up to 180 degrees.  

Using Barbera Rydens method you can see the relations here.

http://cosmology101.wikidot.com/geometry-flrw-metric/

http://cosmology101.wikidot.com/universe-geometry is the first page of the article but the second page given prior is the one showing the universe geometry metrics

Edited by Mordred
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4 hours ago, geordief said:

 So  we can talk about curved 3d space or curved 4d spacetime?

I think the idea @Genady was making is that Galilean gravity wouldn't (or wouldn't have to) be modeled as curvature of anything. It would (or could) just be a regular field in flat space, like the electric field. As I mentioned earlier, the whole point of spacetime and Einstein's work was to make the laws of dynamics and gravity consistent with (Lorentz-invariant) electrodynamics.

Edited by Lorentz Jr
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11 hours ago, geordief said:

 So  we can talk about curved 3d space or curved 4d spacetime?

They are different models.....

I was only familiar with the concept of curved 4d spacetime.

Could you perhaps explain(or give me a pointer)  how  one would measure a curvature  in 3d space .**

I think  I understand  the principle of curvature in 4d spacetime.

 

**I suppose it must refer to the surface of a sphere?

 

 

The concept of Riemann curvature as it is used in ordinary GR applies in any dimension equal to or greater than 2. That means that yes, you can have “curved” 2D and 3D spacetimes as well. The big difference is the level of complexity - in 2D, the Riemann tensor has exactly one independent component, so it is simply a scalar; in 3D, it has 6 components, and can be shown to equal the tank-2 Ricci tensor. Hence these situation have a lot fewer degrees of freedom than we see in our 4D world.

Geometrically speaking, in 2D you have only scalar curvature, so the only thing that can happen is that the area of a 2D surface differs as compared to the same situation in a flat spacetime. It is very simple. In 3D, you get a new kind of curvature which is Ricci curvature, which means that volumes may differ as compared to the same situation in flat spacetime. In 4D and above then you have, in addition to scalar and Ricci curvature, also Weyl curvature, which introduces (relative) tidal forces and shear between neighbouring geodesics, meaning it (roughly speaking) distorts shapes as compared to the flat spacetime situation.

So yes, it is in fact possible and meaningful to talk about GR in three dimensions. However, since such as theory contains only scalar and Ricci curvature, but no Weyl curvature, the resulting phenomenology is very different from what we actually see in the real world - at a minimum there would be no gravitational radiation, no gravity at all in vacuum, and gravity in the interior of bodies would behave quite differently.

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Flat GR does not exactly give Newtonian gravity + Newton's equations + Galilean group (as its ST-symmetry). It produces Newtonian gravity as a theory of the potential + Lorentz-group relativity, which is a strange beast I have never met. In order to obtain Newtonian gravity + Newton's equations + Galilean group, you must substitute the Minkowski metric by the Euclidean metric, while isolating time in so-called simultaneity fibers that don't mix with space under transformations between inertial observers. IOW, time must become absolute.

So I suppose my answer is yes, Galilean gravity is a thing. Galilean group gives you the symmetry group of Newton's equations of motion, while Newton's law of gravitation relates sources of gravitational field with the gravitational field everywhere.

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