The 4D Manifold of Space-time

[geordief]

Preamble: I hope I can post here (in the homework sub forum) since I am trying to "catch up" ** with the subject of Relativity in a way.

 

 

 

If the universe is represented by a model consisting of the set of events in a 4D manifold , are there points in this manifold that have particular characteristics (or properties?)

 

For example ,what does it say if events are separated from each other by the same 4D distance?

 

Secondly the event that represents the Big Bang , would that be the event with the greatest sum of distances from all the other events?

 

Thirdly would this event (the Big Bang) be considered to be at the "centre" of the manifold?

 

Fourthly: Is there a subset of events that have the same "distance" from the Big Bang and what can be said about them that might be interesting?

 

 

 

Can anyone provide me with a link on the internet to where these kinds of questions are answered-or put me right as to whether my questions actually make sense....?

 

Have I badly misunderstood how this manifold model is supposed to work?

 

 

** By "catch up" I mean I am trying to learn some of the standard theory and not to challenge it.

Edited November 8, 2015 by geordief

[studiot]

All of these questions can be answered by understanding what is meant by the '4D distance'

 

Unlike 3D distance the 4D distance has components that add to it and components that subtract from it.

 

In 3D the distance (also called the metric) is given by

 

(d₂ - d₁)² = (x ₂- x₁)² + (y₂-y₁)² + (z₂- z₁)²

 

The things to notice here are that because they are all squares they are all positive and that all the components are added together.

 

So any component value at all will contibute to increasing the distance.

 

In the 4D spactime manifold this is not the case.

 

The distance, now denoted by s, is given by

 

(s₂ - s₁)² = (x ₂- x₁)² + (y₂-y₁)² + (z₂- z₁)² - c²(t₂- t₁)²

 

There is now a term that can subtract from the overal value of distance so a large contribution from the additive components can be offset by a large contribution from the subtractive one and vice versa.

 

Note that this is the standard metric for this work in special relativity, but there are other more complicated ones available that are used in general relativity.

[geordief]

Thanks

Is it possible to set the values of s₁ x₁ y₁ z₁ t₁ to zero?

 

Could those values represent an event at around the Big Bang? and allow us to find the 4D distance of any event with respect to it?

 

 

Am I right that the set of 4D events is increasing or does the fact that the Manifold is described as "static" preclude that possibility?

 

Did my OP "make sense" ? Was it shot through with misunderstandings ?

[studiot]

Thanks

Is it possible to set the values of s₁ x₁ y₁ z₁ t₁ to zero?

 

Could those values represent an event at around the Big Bang? and allow us to find the 4D distance of any event with respect to it?

 

 

Am I right that the set of 4D events is increasing or does the fact that the Manifold is described as "static" preclude that possibility?

 

Did my OP "make sense" ? Was it shot through with misunderstandings ?

 

You can declare any point in the manifold 'zero', but it is arbitrary, just as declaring 0^o longitude to pass through London is arbitrary and Hawaii would have done.

 

This brings up another important point.

About the word curvature.

 

Using the 3D formula I quoted above, all the distance lines between any two points A and B remain in the manifold.

That is they do not pass through points not in the manifold on their 'way' from A to B.

 

If we consider our manifold to be the 2D (surface of the ) sphere with coordinate lines of latitude and longitude then lines, generated by the 3D formula, between points A and B on this surface pass through points not in the 2D manifold.

 

For this to happen we say that the 2D manifold is curved in 3D space and embedded in it.

The alternative is to use more complicated spherical trigonometry (or worse) formulae which keep all lines within the manifold.

 

This is the reason why we need more complicated formulae when we want to discuss the physics of the 4D manifold - we do not want the embedding in a yet higher dimension.

[geordief]

So is the 4D event that describes the Big Bang any different from the other 4D events we can describe mathematically ?

 

Is it just an "unprivileged" member of the set with no way to differentiate it from any of the others?

 

The manifold merely "catalogues" all the events without laying out any "qualitative" differences whatsoever?

 

Is there no kind of kind of "ordering" at all...? (just relationships between arbitrary pairs of events) .

[studiot]

I am not a cosmologist, and have little interest in it.

 

The introduction Minkowski spacetime as a theory predates the big bang as a theory by several decades and I think that the introductuion of the big bang requires severe modification to MST if not something else entirely.

 

There is an interesting mathematical point to be made here.

 

Much of mathematics is based on collecting together into sets objects, points, 'events' or whatever that have common properties.

That is what a manifold is.

A set or collection of points with a specific list of (mathematical) properties.

 

Two properties are desirable for such sets because then we can do extensive mathematics on them

 

1) The set contains all points with the particular property list.

 

2) The mathematical processes we employ (functions, transformations operations whatever) connect one member of the set to another member of the set.

They do not take us outside the set.

 

So adding 3 to 4 produces 7, another number in the set. It does not produce a triangle

 

Processes which lead to a result that is not a point in the set are called singularities (amongst other things).

 

Processess which involve the conjunction of an infinite number of points sometimes lead to a result that is not a member of the set.

For instance Fourier analysis applies to continuous functions in linear analysis, and a fintite combination will always produce another continuous function.

But it is possible to construct discontinuous functions with an infinite number of points.

 

So perhaps the big bang is not a valid member (point) of Minkowski spacetime.

Edited November 8, 2015 by studiot

[geordief]

Thanks.

What if I rephrased it to specify an event "close to" the Big Bang?

 

Is the 4D manifold "blind" to that circumstance?

 

All events are equivalent no matter how close they are the the Big Bang.

 

I think you have answered this studiot when you said

"The introduction Minkowski spacetime as a theory predates the big bang as a theory by several decades and I think that the introductuion of the big bang requires severe modification to MST if not something else entirely."

[studiot]

'close to' is introducing the mathematica (topological)l concepts of connectedness, neighbourhoods, compact sets, all of which I think are violated by the big bang itself.