What is "Geometry"?

[geordief]

Especially in this context....

Mass ,energy etc are said to "curve the geometry of spacetime"

Now I think  I understand that this may be another  way of saying  that ,viewed from a different frame of reference  that "straight lines"  (geodesics) appear to follow a curved trajectory.

If I am right there, what does this say about the  meaning of the word "geometry" ?

Is "geometry"  a formalized setting out of all the different kinds of relationships between  points on a manifold? Would that be an acceptable definition?

I also think I  have learned that all "geometry" is Euclidean at a sufficiently  local level but when the observer tries to encompass wider regions affected by gravity  then it ceases to be so (parallel lines diverge or  meet at infinity)

Is that correct? 

[MigL]

Essentially correct for the GR definition.
Except I would say, it approaches Euclidian at a sufficiently local level.

The mathematical definition will differ.

[geordief]

9 minutes ago, MigL said:

Essentially correct for the GR definition.
Except I would say, it approaches Euclidian at a sufficiently local level.

The mathematical definition will differ.

Thanks.Yes I understand "approaches".

Good to know my understanding is more or less  in accordance with GR.

Can I ,as an analogy (again vis  a vis GR ) maybe understand geometry's relationship to its terrain  like a chamaeleon's to it's?

It swings each and every way...a property of the terrain rather than a thing itself...

[Mordred]

4 hours ago, geordief said:

Especially in this context....

Mass ,energy etc are said to "curve the geometry of spacetime"

Now I think  I understand that this may be another  way of saying  that ,viewed from a different frame of reference  that "straight lines"  (geodesics) appear to follow a curved trajectory.

If I am right there, what does this say about the  meaning of the word "geometry" ?

Is "geometry"  a formalized setting out of all the different kinds of relationships between  points on a manifold? Would that be an acceptable definition?

I also think I  have learned that all "geometry" is Euclidean at a sufficiently  local level but when the observer tries to encompass wider regions affected by gravity  then it ceases to be so (parallel lines diverge or  meet at infinity) 

Is that correct? 

 

 I would like to add some details to the above. You are correct in the above in essence, however there is a more accurate way to think of the above that will help you better understand any field theory than just the spacetime field. First off a field in general is any assigned quantity/function etc assigned under (and this is important) a coordinate basis. In terms of quantity/function the three distinct types are scalar values or any function where the resultant is a scalar value, vector functions which have a magnitude or direction (including spinors) and tensor functions, which the previous aforementioned can also be tensors. (a rank zero tensor is scalar, rank 1 vector).,

 The spacetime metric 4d.. is a rank two tensor. However unless you get heavy into the math the important part is recognizing what defines a field. It is an abstract mathematical construct under a coordinate basis, however not specific to a chosen coordinate type...ie Schwartzchild, Kruskal, Euclid etc. (this is the basis of symmetry relations) for example any two identical vectors in magnitude and angle are symmetric objects at any coordinate. ( in this case identical).

 Now every coordinate is the assigned any of the above, either a scalar, vector or tensor field. In GR every coordinate is called an event.  So really just think of spacetime curvature as a reflection of the values assigned at each coordinate. Now the literal definition of mass under all this is resistance to inertia change ( as per Newtons laws of inertia) in any and all field theories. Energy is just the ability to perform work, neither describe an object but they both describe specifically the field value at a given coordinate location. A large object will have numerous coordinates assigned to it.  As they are also geometric objects ( can be described under a coordinate basis.)

Now GR typically describes the worldlines, geodesics between two coordinates and the ratio of change between them. However this path can be curved. As mentioned mass/energy tells spacetime how to curve, so what does this mean... Well from the above we are probably measuring some set of vectors (ie falling object/ photon light path etc.) we could be using tangent vectors or other. Now in the 4D spacetime, time is treated as another coordinate axis. (key note time under GR is both coordinate time and proper time. ) The tau symbol [latex] \tau[/latex] being proper time

https://en.wikipedia.org/wiki/Proper_time

https://en.wikipedia.org/wiki/Coordinate_time

as you can see from the links each event is described by coordinate time, now under SR this is the at rest frame described as Euclid as it follows Newton's laws of inertia to good approximation our everyday existence. The proper-time will follow either a curved or straight path. Its important to note that it is the proper-time that follows the geodesic line however at each coordinate (event/reference frame) the observer will observe a Euclidean geometry. It is when you compare two events/reference frames that a distinction becomes apparent.

 Hope this helps you take the next step into SR/GR or any field theory in general.

 Always keep in mind, spacetime curvature etc is a mathematical representation of change of a volume being measured or compared. Terms such as fabric/ manifold/fire bundles etc are simply descriptive assigned terms to help describe math relations between those field values.. LOL that list includes strings in string theory...ie the string is some waveform being described. Ratio of change between coordinate and proper time being an example of change occurring. Described as a geometry change.

 

 

Edited January 16, 2019 by Mordred

[studiot]

4 hours ago, Mordred said:

First off a field in general is any assigned quantity/function etc assigned under (and this is important) a coordinate basis.

 

Yes this is the most important bit.

 

What it is telling you is that all the posh language and fancy terms are really talking about a glorified graph. Just like we drew in junior school, but perhaps a bit more complicated.

All the fancy maths is doing is enabling us to pick out or read values from this graph.

This is an entirely geometric process in the junior school meaning of the word geometry, and is used by millions of Engineers every day for this purpose.

 

 

7 hours ago, MigL said:

Essentially correct for the GR definition.
Except I would say, it approaches Euclidian at a sufficiently local level.

The mathematical definition will differ.

I fully agree. Short and sweet and to the two points. +1

Here this is perhaps the most important bit.

"The mathematical definition will differ."

Mordred was offering the GR definition as used in GR, (which is also my glorified graph version )

The mathematical viewpoint is entirely different.

Edited January 16, 2019 by studiot

[MigL]

So, are you going to tell us the mathematical definition, or keep us in suspense ?