The Fourth Dimension: Introduction to a Book

[DaveC426913]

What two objects?

You measure an object here, and then you measure an object a light year away. But then you discover there's a third object, another light year away, so you measure it. In doing so, you discover another object another ly away. Every time you go to measure an object, you discover another object. You never come to the end of objects.

Where is the centre of a line of infinite length?

 

[Tac-Tics]

Where is the centre of a line of infinite length?

0.

=-P

You measure an object here, and then you measure an object a light year away. But then you discover there's a third object, another light year away, so you measure it. In doing so, you discover another object another ly away. Every time you go to measure an object, you discover another object. You never come to the end of objects.

So it seems your argument is that we don't have perfect knowledge of all massive objects.

But let's suppose the universe on such a large scale is classical enough for objects to have definite mass and position independently of measurement. Furthermore than the number of massive objects is finite. And that the universe isn't "well balanced" such that each pair of objects has a unique shortest path between them. Then, there should be a center of the universe, even if it is beyond our ability to prove where it is.

 

[stealthtank91]

What two objects?

You measure an object here, and then you measure an object a light year away. But then you discover there's a third object, another light year away, so you measure it. In doing so, you discover another object another ly away. Every time you go to measure an object, you discover another object. You never come to the end of objects.

Where is the centre of a line of infinite length?

yes, but the scenario your describing has an infinite amount of objects at approximately 1 ly away from each adjacent other one. but Einsteins classic formula e=mc^2 proves that there is a limit of matter in our universal space or else we could travel at the speed of light. but the point is that there is a certain amount of this matter, and there HAS to be a average center of the overall amount.

however i do understand and agree with your reasoning on the boundless universe concept. "if there is no edge there can't be a center"

 

[DaveC426913]

Again, I refer you to the balloon analogy. The balloon has a finite mass and a finite number of objects; it has objects a finite distance on the surface from each other. Where, on the surface of the balloon, would you be able to point to and say "there is the centre of mass."

 

[DaveC426913]

As a sidenote: 2 non sequiturs in the same sentence.

e=mc^2 proves that there is a limit of matter in our universal space

this does not follow

there is a limit of matter in our universal space or else we could travel at the speed of light.

this does not follow

 

[Tac-Tics]

As a sidenote: 2 non sequiturs in the same sentence.

this does not follow

this does not follow

I totally agree that those statements do not follow, so please do not misquote me. I was not the one who said either of those things.

Anyway

Again, I refer you to the balloon analogy. The balloon has a finite mass and a finite number of objects; it has objects a finite distance on the surface from each other. Where, on the surface of the ballon, would you be able to point to and say "there is the centre of mass."

Let's do a simple two dimensional example of objects of equal mass on the perimeter of a circle. Let's say we have three masses, m1, m2, m3 each at their respective position given by angular coordinates 0, pi/3 and pi/2.

To average the objects, we take the location of each (their angle coordinate), multiply it by their mass (we assume to be a unit here), and divide by the total mass (which will be 3 units). So the average position of these would be (0 + pi/2 + pi/3) / 3 = 5 pi / 18. That gives us a definite location on the circle which might aptly be named a "center" of the circle weighted by mass.

 

[Randomguy]

^I think you're forgetting that the Universe is isotropic in nature. Which means that every point in the Universe is the centre, or in other words, no point is the centre.

If we were to do what you suggest, insofar as I know, we would calculate that the Earth is the centre of the Universe. However, if we moved to another point in the Universe and calculated where the centre was, we would find that that too was the centre.

To see the flaw in the idea, try answering this. Where is the centre of the surface of a sphere?

Obviously the surface of a sphere has no centre, and this is analogous to the Universe.

 

[Tac-Tics]

^I think you're forgetting that the Universe is isotropic in nature. Which means that every point in the Universe is the centre, or in other words, no point is the centre.

What exactly do you mean by isotropic in this context and what theory does this idea stem from?

To see the flaw in the idea, try answering this. Where is the centre of the surface of a sphere?

Obviously the surface of a sphere has no centre, and this is analogous to the Universe.

The geometric center of a space would be the average of all points in the space equally weighted. I'm sure you could find a calculus-based interpretation that gives this a full and meaningful definition. In this sense, the set [0, 1] in R would have a "center" of 1/2, the center of a closed unit ball in R^n would be 0, and spaces like R^n or the boundary of a ball would be centerless either because the center would be indeterminate (such as R^2) or because the center is not a point in the space (such as the boundary of a ball).

But you could define other "centers" where the weights of each point are distributed in a non-uniform fashion. In one dimension, this would essentially be a simple probability distribution, and the center would be the mean. This distribution could be generated by physical properties of each object in space, such as mass.

My question is could there not exist such an alternative "center" that is well-agreed upon?

 

[Randomguy]

What exactly do you mean by isotropic in this context and what theory does this idea stem from?

It stems from the Cosmological Principle. (aka Copernican Principle)

http://en.wikipedia.org/wiki/Copernican_principle

It states that the Universe is homogeneous and isotropic. This is not fact, but there is certainly good evidence for the isotropy of the Universe (uniformity of cosmic microwave background radiation).

http://en.wikipedia.org/wiki/Isotropic

The geometric center of a space would be the average of all points in the space equally weighted. I'm sure you could find a calculus-based interpretation that gives this a full and meaningful definition. In this sense, the set [0, 1] in R would have a "center" of 1/2, the center of a closed unit ball in R^n would be 0, and spaces like R^n or the boundary of a ball would be centerless either because the center would be indeterminate (such as R^2) or because the center is not a point in the space (such as the boundary of a ball).

I haven't really done that before, but from what I understand the Universe has no geometric centre. It cannot because it does not have boundaries.

But you could define other "centers" where the weights of each point are distributed in a non-uniform fashion. In one dimension, this would essentially be a simple probability distribution, and the center would be the mean. This distribution could be generated by physical properties of each object in space, such as mass.

If you take this approach, according to current understanding you would still not find a centre of mass, because the Universe is homogeneous on macroscopic scales.

My question is could there not exist such an alternative "center" that is well-agreed upon?

I suppose it could, but you'd probably need someone more knowledgeable to answer that. However, due to the generally believed homogenuity of the Universe it shouldn't have one. But if there was a region with an extremely high concentration of matter, it could perhaps be viewed as the centre...

 

[DaveC426913]

But you could define other "centers" where the weights of each point are distributed in a non-uniform fashion. In one dimension, this would essentially be a simple probability distribution, and the center would be the mean.?

Again, where is the centre of mass of masses distributed along an infinitely long one-dimensional line? Whereever you stop to measure, you find you're missing some masses just beyond where you stopped. The only way to define a centre is to have ends.

This distribution could be generated by physical properties of each object in space, such as mass.

My question is could there not exist such an alternative "center" that is well-agreed upon?

But if there was a region with an extremely high concentration of matter, it could perhaps be viewed as the centre...

If you glued one hundred pennies on a basketball, and then noticed that some of them clustered in one place more than elsewhere, you could, if you so chose, "define" that as the "centre" of the surface of the basketball. And it would be as useful and meaningful a definition as that of defining the universe's centre by some concentration of mass.

i.e.: not.

 

[Tac-Tics]

If you glued one hundred pennies on a basketball, and then noticed that some of them clustered in one place more than elsewhere, you could, if you so chose, "define" that as the "centre" of the surface of the basketball. And it would be as useful and meaningful a definition as that of defining the universe's centre by some concentration of mass.

i.e.: not.

The referees employed by our friends at the NBA might disagree. The cluster point of pennies on a basket ball would be an important constant influencing the motion of the ball.

Maybe it's questionable whether or not this kind of center would be meaningful, but it seems like it lends itself to an unbiased coordinate system in the universe.

I'm still interested, though, in how this artificial center would be influenced by relativity. If everything is standing still, you can be completely objective about it, but I'm sure that all goes to hell when you lose simultaneity.

 

[Randomguy]

^I don't think the impact of having a 'centre' of mass would be all that great. I mean, it would just be a more exaggerated effect of what's already happening in dense parts of the Universe (such as galaxies). i.e. The Universe in dense areas would not expand because of the gravitational pull, but the Universe in the surrounding less dense areas would expand rapidly.

 

[DRMOKADI]

But theoretically its known that the Earth is moving around the axis. This means that the axis support the Earth at its centre.

 

[mysearch]

This thread touches on many interesting issues, so I wanted to see if there was any consolidation of opinion on a number of the key issues discussed:

1) N-spatial dimensions:
While the analogy of flatlander and 4D animals are useful visualisations, do they really provide any meaningful picture? While mathematics is free to conceptually imagine any number of spatial dimensions is there any empirical evidence to support a physical existence? The only example I can think of relates to string theory, but I understand that most of these dimensions would have to exist on a quantum scale?

think the truth is that there is no such thing as time, only space. Time is an illusion created by our limited perception.

In post #18, it was suggested that time was only a manifestation of the mind. This was backed up by an example of a pencil moving from A to B and therefore, in this definition of 4D spacetime, simultaneously existing at all points between A & B. While this may be the case, it seems to require a new science in the sense that another pencil moving from C to D, but intersecting the path AB, would violate Pauli’s exclusion principle?

2) 4D Spacetime
As I understand it, the definition of 4D spacetime, as used in relativity and cosmology, is typically referring to 3 spatial dimensions and 1 time dimension. Therefore, while accepting that 4D spacetime is a useful concept for explaining time dilation and space contraction in terms of the spacetime interval, it would seem that an intuitive distinction between time and space can still be retained?

3) Geometry of Space and Time
In the context of general relativity, the Schwarzschild metric suggests that the perception of both space and time can be affected by the presence of a large gravitational mass. However, these effects are only observed at a distance and not by whose within the observed frame of reference. Equally, the more dramatic effects of this form of spacetime curvature are typically localised to points of extreme gravitational mass, e.g. black holes. As such, we seem to require a centre of mass or centre of energy density?

In contrast, the FRW metric, based on the assumptions of a cosmological model being both homogeneous and isotropic, seems to reflect that only space expands as a function of time as defined by the scale factor a(t). As such, this leads to a geodesic nature of spacetime, which might be visualised in terms of 2 photons moving in parallel, 1 metre apart. After a period of time, the original separation must be subject to expansion and therefore our parallel photons follow a geodesic and not a classical straight line. As such, this seems to be the definition of spacetime curvature?

Again, while accepting that it might be convenient to fully embrace spacetime as a single entity, it is not clear to me that space and time do not remain different and distinct concepts?

4) The Geometry of the Universe
Within cosmology, there is an additional definition of spatial curvature [k]. With reference to Friedmann’s equation, [k] can be defined as an energy density, which is inversely proportional to the square of the scale factor a(t), i.e. 1/a^2. However, based on this definition, the effects of spatial curvature will have been swamped by matter and radiation in the past and by dark energy in the future. Overall, the current model suggests that [k] reflects an essentially flat spatial geometry.

Therefore, while the balloon analogy might be a useful visualisation of an expanding universe, it is unclear to me how the geometry of the physical universe can be liken to the curved surface of the balloon?

Many of the posts seem to be quite adamant about the issue of the universe having no edge or centre. While taking no issue with this position within the context of the standard model, this does not automatically preclude other models leading to other conclusions. I am not forwarding the following model in opposition to the standard model, see link below, but some of the readers of this thread might be interested in some of the ideas expressed by people with some obvious academic background. http://arxiv.org/abs/gr-qc/0602102

Would be interested in any clarifications or additional comments on any of the points raised. Thanks

 

[Randomguy]

@DRMOKADI: Sorry, I don't follow your point. What axis are you talking about?

If you're talking about the axis the Earth revolves around, that is irrelevant on a Universal scale. Each planet revolves around its own axis...

The only example I can think of relates to string theory, but I understand that most of these dimensions would have to exist on a quantum scale?

Exactly.

While this may be the case, it seems to require a new science in the sense that another pencil moving from C to D, but intersecting the path AB, would violate Pauli’s exclusion principle?

The Pauli exclusion principle is only with respect to time, so no, I do not believe it would violate it.

Therefore, while accepting that 4D spacetime is a useful concept for explaining time dilation and space contraction in terms of the spacetime interval, it would seem that an intuitive distinction between time and space can still be retained?

I'm not really sure what you mean, but I don't think you can maintain as clear a distinction between them as was previously thought by Newton etc. Clearly one affects the other, but I'd agree they aren't the same.

But don't ask me, I'm only a 17 year old. :p

Therefore, while the balloon analogy might be a useful visualisation of an expanding universe, it is unclear to me how the geometry of the physical universe can be liken to the curved surface of the balloon?

I don't believe the balloon analogy should be taken too far to apply to the shape of the Universe. I think it is only used to explain how Earth is not in the centre despite everything moving away from it.

Anyway, I tried to answer the bits I could try to answer.

 

[mysearch]

The Pauli exclusion principle is only with respect to time, so no, I do not believe it would violate it.

Fair point, but the argument seemingly being put forward in #18 was that “there is no such thing as time, only space”. Therefore, I was only highlighting that removing the concept of time has knock-on effects with respect to current science.

I'm not really sure what you mean, but I don't think you can maintain as clear a distinction between them as was previously thought by Newton etc. Clearly one affects the other, but I'd agree they aren't the same.

As far as I am aware, the main difference is that Newton saw time as an absolute concept, while Einstein describes time as a relative concept. Therefore, I was just questioning whether the concept of spacetime really affects our separate perception of space and time.

I don't believe the balloon analogy should be taken too far to apply to the shape of the Universe. I think it is only used to explain how Earth is not in the centre despite everything moving away from it.

I agree. However, to many it seems to infer some description of a boundless universe. Therefore, I was just curious to known whether there was any rationale behind this inference or that it was simply pushing the analogy too far.

But don't ask me, I'm only a 17 year old.

Unfortunately age isn’t always an indicator of intelligence or wisdom. If it were I would be a lot smarter.

 

[maxvan]

I think there doesn't exist a four dimension. Although visual spatial learner can see with 3 dimension in their mind. so I guess what you call the fourth dimension is actually the 3 dimension but spatial. just like you have 3 d en ddd stands for 3 d such like windows vista and mac, the screen can turn itself, unlike the ddd that de picture can come out of the screen instead, and that create the spatial effect. I am a visual spatial learner, and see picture and movies in 3d en ddd (spatial) in my mind. I know exactely where is something wrong.

 

[JANm]

Dear Maxvan
Allright so you have a 3-D imagination and also some beginning technology on that area. I have some ideas how to make the fourth vivid.
1 A ship or a duck makes a fore in the water. It is adding up circles; and if the ship/duck bathe in this situation for a conciderable time then it tends to a V,
but what if the ship/duck just started?
2 An airplane drops parachutists. You see the first perhaps with open shute and others with a downgoing line ending in the open door of the plane.

are those four dimensional pictures or not?
greetings JnM