Theories on space

[gre]

Are there any theories out there that consider space a field (or Field-Like? If space can't exist without matter and vis versa, wouldn't that imply they are somehow tied together?

 

If a single particle/mass existed in an area without space, maybe a specific quantity of space would exist around that particle/mass?

[ajb]

There are arguments that suggest that space-time could be emergent. That is they are not fundamental in the theory but come as a macroscopic approximation.

 

Theories based on noncommuative geometry and limits of string theory exist, one difficult that I am aware of is are the non-local properties.

 

Nathan Seiberg summarszed some of these ideas in "Emergent Spacetime" arXiv:hep-th/0601234v1

[Galindo]

Space=Time

[ajb]

Space=Time

 

Indeed, this should be the first thing one learns in relativity 101.

 

As we have transformations that mix space and time it is not clear in general how to make a distinction between space and time.

[michel123456]

Indeed, this should be the first thing one learns in relativity 101.

 

As we have transformations that mix space and time it is not clear in general how to make a distinction between space and time.

 

"Space=Time"

I am glad to see that writed down by someone (Galindo) and supported by AJB. Usually when i go on such an assertion I am treated like a troll and being presented a list of major differences between space & time.

 

For example, when discussing the concept of "the arrow of time", people use to oppose that there is no such a concept applying to space. In space, they say, there is no "arrow", you can go everywhere.

And like a troll, I begin into disagreeing with this "fact".

And try to explain that there is an "arrow" in space too.

Usually I am ejected at once.

 

Now that AJB seems to back up the space=time equation, can I try again explaining my ideas?

Edited February 3, 2010 by michel123456

[ajb]

The thing is I know what is meant by "space = time".

 

The ethos of general relativity and indeed its mathematical formulation is that there is in general no canonical way to separate space-time into space and time.

[gre]

What are some supporting arguments/facts for "space = time" (from general relativity, etc)?

[ajb]

What are some supporting arguments/facts for "space = time" (from general relativity, etc)?

 

Think about special relativity. Let us pick inertial coordinates [math]\{t,x,y,z\}[/math]. Now, we are free to pick any other system of inertial coordinates. These are related by a (passive) Lorentz transformation. (not worry about the affine bit). If you look at the Lorentz transformation to a frame [math]\{t',x', y, z\}[/math]you see that the [math]t' = t'(t,x)[/math] and [math]x'= x'(t,x)[/math]. So we have transformations that mix space and time.

 

As no particular inertial reference frame is preferred we conclude that one cannot canonically separate space-time into space and time.

[Klaynos]

In special relativity it can be shown, using the 4 vector, that the 3 space and 1 time dimensions 'rotate' so that what looks space like in one reference frame looks time like in another.

[michel123456]

Speaking technically, in the set of inertial coordinates [math]\{t,x,y,z\}[/math] only [math]\{t\}[/math] is always positive, negative time does not exist.

By opposition [math]\{x,y,z\}[/math] can be both positive or negative.

How come that something always positive by definition can be transformed in something that can change sign?

 

In other words, if time can be changed in space and vice-versa, what about the arrow of time?

You all should agree that if the arrow exist in time, it must also exist in space, don't you?

Where is it?

Edited February 4, 2010 by michel123456

[ajb]

The t coordinate should not really be thought of a time in the sense you mean. Initially, think of it as another coordinate, the physical interpretation will come later. It is not necessarily positive.

 

The proper time measured along a time-like path is positive. This you cannot take to define an arrow of time.

[michel123456]

It is not necessarily positive.

 

So [math]\{t\}[/math] considered as a coordinate can be positive or negative.

I suppose the physical interpretation is:

positive = past

negative = future.

Correct?

[ajb]

Yes.

 

You should think about things in terms of a Minkowski diagram. (Even better in in general relativity is a Penrose diagram).

[michel123456]

Yes.

 

You should think about things in terms of a Minkowski diagram. (Even better in in general relativity is a Penrose diagram).

 

Ha.

So, in a Minkowski diagram, I can put a + sign in the low part (the past) and a minus sign in the upper part (the future).

 

Edited February 4, 2010 by michel123456

[ajb]

That is ok.

 

Consider the space-time interval

 

[math]ds^{2} = - (dt)^{2} + (dx)^{2}[/math]

 

as defined in our original coordinates.

 

Then consider the transformation [math]t \rightarrow t' = -t[/math]. Then [math]dt \rightarrow dt' = - dt[/math] and this the space-time interval is invariant under such a change of coordinates.

 

You should now look up T-symmetry.

[michel123456]

Good.

What a relief.

 

T symmetry is related to that weird thing that happen in physical real life, this arrow represented by the blue arrow on the diagram. The blue arrow of time is introduced arbitrarily, i.e. because we know that's the way things happen, and not because it comes from some physical law.

But instead of looking at T symmetry, I'd like to look at space.

 

We know that time is always positive, but we can have positive & negative coordinates for time.

Regarding space, positive & negative coordinates are of common usage too.

But there is something that is always positive in space.

Distance.

Negative distance does not exist, as negative time does not exist.

And you know that distance can be expressed as time, and vice-versa.

Correct?

[ajb]

As the metrics in relativity are not positive definite there is a notion of "negative distance" and "zero distance" between (near by) distinct points. Look up time like, space like and light/null like paths/vectors.

 

The transformation [math]x^{\mu} \rightarrow -x^{\mu}[/math] is known as a parity transformation. You see that the space-time interval is also invariant under such transformations. (You can also consider a parity shift on each spacial coordinate separately).

 

Now look up P-symmetry and PT-symmetry.

 

Something that is not always made clear is that one must be careful when talking about symmetries in general. The laws can be symmetric or not and separately the states can be symmetric or not.

[ponderer]

Space=Time

 

I thought time=money.

 

I am starting to develop this idea of space-time-energy,

 

They go together like a horse and carriage (and harness maybe)

 

you can't have one,

you can't have one,

you can't have one,

without the ooo--ooo-oother.

 

You cannot demonstrate time without space and energy

You cannot demonstrate space without energy and time

You cannot demonstrate energy without space and time

Edited February 5, 2010 by ponderer

[ajb]

I am starting to develop this idea of space-time-energy,

 

You should probably also include momentum. Look up the various extended phase spaces that can be found in nonautonomous classical mechanics. As you know momentum is the conjugate coordinate to position, well you can follow this through and realise that energy is conjugate to time. You can then more or lass formulate classical mechanics as you would in terms of symplectic geometry for autonomous mechanics.

 

Related to this is multisymplectic geometry and multisymplectic field theory.