Consequences or not for a Geometrical interpretation of GR

[geordief]

I am learning that AE never accommodated himself to a geometrical interpretation of GR (stand to be corrected of course)

 

Do any practical consequences flow from the adoption of his interpretation of GR as against what I understand is now generally accepted as the "best" interpretation.

Or is it just you say "tomato" and I say "tomato"?

 

For instance ,would it matter when considering any of the possible theories around Quantum Gravity  whether Einstein was right or wrong? 

[Strange]

18 minutes ago, geordief said:

I am learning that AE never accommodated himself to a geometrical interpretation of GR (stand to be corrected of course)

Where does that come from? I don't think it can be correct. After all, SR is a purely geometrical description of the relation between space and time. Einsteins started by considering SR under conditions of acceleration. 

Then he considered what would be observed by an observer on a rotation disk (for example, they would measure a different value for pi) and so he realised he needed to consider non-Euclidean (curved) geometry.  

His reluctance here may have been that the mathematics is very complex and (despite the fact he was not the mathematical dummy some claim) he found it daunting. He then worked with a friend, Marcel Grossman, to learn about differential geometry and tensors.

More here: https://en.wikipedia.org/wiki/History_of_general_relativity

 

[geordief]

2 hours ago, Strange said:

Where does that come from? I don't think it can be correct. After all, SR is a purely geometrical description of the relation between space and time. Einsteins started by considering SR under conditions of acceleration. 

Then he considered what would be observed by an observer on a rotation disk (for example, they would measure a different value for pi) and so he realised he needed to consider non-Euclidean (curved) geometry.  

His reluctance here may have been that the mathematics is very complex and (despite the fact he was not the mathematical dummy some claim) he found it daunting. He then worked with a friend, Marcel Grossman, to learn about differential geometry and tensors.

More here: https://en.wikipedia.org/wiki/History_of_general_relativity

 

Here is where I  found (and participated in) that discussion.

 

https://www.thenakedscientists.com/forum/index.php?topic=65940.msg556162#msg556162

[Strange]

I don't really understand what the comment you linked to is trying to say.

In the later comment where Einstein is quoted as saying:

Quote

I do not agree with the idea that general relativity is geomterizing physics of a gravitational field. 

Someone seems to have misinterpreted this. He isn't saying that it's not geometrical; he is saying it is no more geometrical than any other physical theory (which is entirely true: you can express electromagnetism in geometrical terms, for example).

Edited October 12, 2018 by Strange
formatting

[MigL]

GR can be interpreted as a field theory where geometry is the field.
 What is the distinction ?

But I see the point about gravity being an apparent force ( inertial or fictitious like centrifugal or Coriolis ) , as it is only manifest in certain types of frames of reference.

[Markus Hanke]

10 hours ago, MigL said:

What is the distinction ?

Distinction between what?

[MigL]

As relating to the OP, Markus.
Between a geometric theory and a field theory.

[geordief]

23 hours ago, MigL said:

GR can be interpreted as a field theory where geometry is the field.
 ?

I don't understand  your phrase ,"geometry is the field" .

 

Do you mean there is a  map of a set of points (events)  where each point has a different geometry (spacetime curvature)?

 

Spacetime curvature represents  a kind of geometry?

 

 

[studiot]

Geordief

There seems to be some doubt as to what you understand by the phrase geometrical interpretation and the word geomterizing .

Look at these two plots from thermodynamics

One is a 3D  plot of the Van der Waals equation of state.

The other is a 3D plot of the states of a chemical reaction.

Wouuld these conform to your idea of geomterizing  ?

 

 

[MigL]

A field is simply a value associated with each point.
That value can be scalar, vector or tensor.

For GR the value at each point is a measure of the deviation from 'flat'.
IOW the field is geometric.

[geordief]

26 minutes ago, studiot said:

Geordief

There seems to be some doubt as to what you understand by the phrase geometrical interpretation and the word geomterizing .

Look at these two plots from thermodynamics

One is a 3D  plot of the Van der Waals equation of state.

The other is a 3D plot of the states of a chemical reaction.

Wouuld these conform to your idea of geomterizing  ?

 

 

 

 

I should be well chuffed to have been apparently  mistaken for Einstein as That was Einstein's quote and not mine(Strange copied the Einstein quote from the link I provided on the other science forum ;it is the 4th post in the thread)

 

I don't know if I should attempt to answer your question  as I too am trying too understand that Einstein quote (amongst other things)

Edited October 13, 2018 by geordief

[studiot]

I haven't many more time for some hours tonight but the idea is to get you thinking of the context in which that Einstein's comment was made.

 

Concentrate on the Van der Waals plot.

(Look up VdW if you need to)

The ideal gas equation is PV = RT connecting pressure (P) Volume (V) and temperature (T) via the gas constant.

A very simple equation and if you plot lines of constant volume you get a Euclidian grid (try it)

P = a constant times T.

The dieal gas law assumes that the gas molecules occuply no volume.

Real gases are made up of molecules that take up some volume - different for different gases.

VdW introduces a correction to the ideal law.

This correction distorts the regular euclidain grid, bringing some lines closer together concentrating them.

 

 

In the same way space with no mass in it is euclidian.

Introducing mass according to GR brings some gridlines closer together, concentrating them.

In both cases the concentration produces curvature of the gridlines.

 

The gridlines themselves are artificial and occur in what we call phase space.
The measure some property of interest.
 

 

[geordief]

4 hours ago, studiot said:

I haven't many more time for some hours tonight but the idea is to get you thinking of the context in which that Einstein's comment was made.

 

Concentrate on the Van der Waals plot.

(Look up VdW if you need to)

The ideal gas equation is PV = RT connecting pressure (P) Volume (V) and temperature (T) via the gas constant.

A very simple equation and if you plot lines of constant volume you get a Euclidian grid (try it)

P = a constant times T.

The dieal gas law assumes that the gas molecules occuply no volume.

Real gases are made up of molecules that take up some volume - different for different gases.

VdW introduces a correction to the ideal law.

This correction distorts the regular euclidain grid, bringing some lines closer together concentrating them.

 

 

In the same way space with no mass in it is euclidian.

Introducing mass according to GR brings some gridlines closer together, concentrating them.

In both cases the concentration produces curvature of the gridlines.

 

The gridlines themselves are artificial and occur in what we call phase space.
The measure some property of interest.
 

 

If I have understood the analogy ,the volume of the molecules locally displace the  medium (the medium just being the volume measurements and so being abstract-a bit like a hole in the ground changing the geometry of the surrounding space?)

 

Is there any corresponding model whereby mass could  displace the thing it is "embedded" in? The Higgs Field perhaps?

 

Apologies for the probable evidence of  obtuseness ,I am learning to live with it (have always been a very slow learner and am getting worse) 

 

Btw I have accepted Strange's clarification of AE's stance on geometrization (it is wrong to think he was objecting to some unique geometrizing quality of GR )

[studiot]

On 14/10/2018 at 11:30 AM, geordief said:

If I have understood the analogy ,the volume of the molecules locally displace the  medium (the medium just being the volume measurements and so being abstract-a bit like a hole in the ground changing the geometry of the surrounding space?)

 

Is there any corresponding model whereby mass could  displace the thing it is "embedded" in? The Higgs Field perhaps?

 

Apologies for the probable evidence of  obtuseness ,I am learning to live with it (have always been a very slow learner and am getting worse) 

 

Btw I have accepted Strange's clarification of AE's stance on geometrization (it is wrong to think he was objecting to some unique geometrizing quality of GR )

 

On 13/10/2018 at 8:37 PM, studiot said:

A very simple equation and if you plot lines of constant volume you get a Euclidian grid (try it)

Did you try the plot I suggest?  It takes all of 15 seconds to sketch.

 

No I am not talking about 'replacement' or 'displacement' or 'embedding'.

 

My analogy is simply offering the idea that geometrisation is another word for plotting or drawing a (multidimensional) graph using suitable axes.

 

So you given two of the three variables, pressure, temperature and volume you can either

 

Calculate the third from the equation

or

Read of the value of the third from the plot.

 

The simple ideal gas law equation and plot are regula, (isotropic and homogeneous)

The more complicated VdW equation is not.

[geordief]

5 minutes ago, studiot said:

 

Did you try the plot I suggest?  It takes all of 15 seconds to sketch.

 

No I am not talking about 'replacement' or 'displacement' or 'embedding'.

 

My analogy is simply offering the idea that geometrisation is another word for plotting or drawing a (multidimensional) graph using suitable axes.

 

So you given two of the three variables, pressure, temperature and volume you can either

 

Calculate the third from the equation

or

Read of the value of the third from the plot.

 

The simple ideal gas law equation and plot are regula, (isotropic and homogeneous)

The more complicated VdW equation is not.

No I didn't try it physically . It just seemed obvious.;P=kT  where k is a constant  caused by the combination of whatever value of constant Volume is chosen  and R ,which is another constant I am not familiar with  but accept(the Gas constant)**

 

It is just the same as a two dimensional  y=ax graph ,ain't it?

 

Are you just saying that all sets of mathematical values can be represented geometrically and so to describe them as being "geometrical" it only trivially true?

 

**Also I don't know how to upload any of my drawings except as a photo which gives me headaches (not seriously)

 

 

[studiot]

9 minutes ago, geordief said:

It is just the same as a two dimensional  y=ax graph ,ain't it?

Yes but a whole family of them for different values of a, describing a disk centered at the origin - in other words a euclidian plane.

Stack these up (for different values of V) and you will get a cylinder, ie a euclidian 3D space.

12 minutes ago, geordief said:

Are you just saying that all sets of mathematical values can be represented geometrically and so to describe them as being "geometrical" it only trivially true?

Well yes but not all of such sets, only  ones which can be described by the property of continuity - the integers cannot for instance.

That is where continuity comes in and topology is sometimes describes as the mathematics of continuity

 

[geordief]

26 minutes ago, studiot said:

Yes but a whole family of them for different values of a, describing a disk centered at the origin - in other words a euclidian plane.

Stack these up (for different values of V) and you will get a cylinder, ie a euclidian 3D space.

 

 

I think you were also telling me that this 3d euclidean cylinder is stressed out of shape by the presence of molecules/atoms  within the physical  structure.

 

A bit like a  tumour  in the human body,

 

Is it also possible to  have "holes" in the 3d cylinder that would model other physical processes? (where the stress works inwards rather than outwards.......)

 

Edited October 15, 2018 by geordief

[studiot]

7 hours ago, geordief said:

I think you were also telling me that this 3d euclidean cylinder is stressed out of shape by the presence of molecules/atoms  within the physical  structure.

 

A bit like a  tumour  in the human body,

 

Is it also possible to  have "holes" in the 3d cylinder that would model other physical processes? (where the stress works inwards rather than outwards.......)

 

Only if you use the more advanced or exact VdW equation.

 

8 hours ago, studiot said:

No I am not talking about 'replacement' or 'displacement' or 'embedding'.

 

 

[geordief]

1 hour ago, studiot said:

Only if you use the more advanced or exact VdW equation.

 

 

 

Trying to orient my understanding of this Euclidean 3d space as against the distorted version which more accurately represents the reality one is interested in can I ask what ,if anything  the geodesics can be used to represent?

There is no analogy to spacetime   geodesics  ,is there? Nothing moves along them ,does it? 

[studiot]

Actually they do have a meaning.

For instance movement along lines of constant volume involve zero work in the First Law of Thermodynamics.

 

But it is best not to push the analogy of least mechanical work lines in mechancal theory (geodesics) and zero work lines in Thermo too far.

 

The point is that both are plots or graphs of properties of interest.

And this connection is geometric.

 

But there is no guarantee of direct correspondence between the properties.