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can you visualize Past-Light-Cones in hyperspace?


Widdekind
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Can the fabric of space-time be considered, to be 'embedded', in a higher-dimensional 'hyper-space' ?

 

For example, in 'flat' space-time, the 'past light-cones' (PLC), of an observer, at some particular 'event' (x,t), are some sort of "hyper-stack" of spherical shells, into some sort of "sphere-cone" hyper-shape -- a little like a concentric set of onion shells, that have been "pulled out" along the time axis:

 

lightconeflat.jpg

In 'closed-but-static' space-time, the "hyper-stack", of spherical shells, comprising the PLC, both "grows & shrinks" in size; and, "porpoises up-and-down through hyperspace" (as the light-rays bounding the PLC "spiral around" the "hyper-cylinder" space-time fabric):

 

lightconecylinder.jpg

In "closed-and-expanding" space-time, the PLC "grows-and-shrinks"; and, "porpoises" as above. However, looking backwards in time, the contracting size-scale of the space-time fabric imposes an "over-all shrinking" on the sizes, of the otherwise "growing-and-shrinking" spherical shells, "hyper-stacked" into the PLC:

 

lightconecone.jpg

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Beautiful diagrams but I understand nothing, maybe you are making it more complicated than necessary.

 

At your question

can you visualize Past-Light-Cones in hyperspace?

The answer is yes.

In space, at a given time T1, the so-called light-cone resumes to the surface of a sphere of radius T1 around you. You being the observer. And radius being expressed in measure of time by C, C being the Speed Of Light.

At time T2, the surface of the sphere expands at radius T2

At T3, radius becomes T3.

And so on.

So the more you look far away, the more T is big, and the more huge is the surface of the sphere.

One can thus visualize this as a sum of ever growing spheres. You can put all those spheres next to each other, so that you make a kind of pyramid made of balls. Then desinflate the balls that become flat as a disk, and you get something like a spacetime diagram.

In reality, all those spheres are the one inside the other, centered on you the observer.

Edited by michel123456
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Don't really understand what you are getting at with the light cones, but I have various objections to the statement

 

"Can the fabric of space-time be considered, to be 'embedded', in a higher-dimensional 'hyper-space' ?"

 

I would have to say no.

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Yes, the fabric of spacetime can be thought of as embedded in a higher dimesional space. It is called an embedding diagram. For example, see link:

 

http://www.astrosoci...2/lockwood.html

 

Here's how I think of it. Take two points in empty space. They are a certain distance apart.

 

Let's place the Sun near the two points (so that a line between the two points goes through the center of the Sun -- i.e. a radial line). Now the distancee between the two points is stretched by the presence of the Sun's mass/energy. How do you depict this longer distance? One way is to draw a downward curve between the two points (or upward) representing the longer distance. This curve is extended in a hypothetical space called hyper space.

post-31036-0-01413200-1323127851_thumb.jpg

Edited by IM Egdall
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So this dimension in which our space is embedded in, did it begin with the big bang as our space did or was it always there ?

Your 'embedding' diagram deals only with the three spatial dimensions, what about time, does it continue to the 'external dimension ?

I myself don't see how it could since space and time are part of the greater whole, space/time, and as such are linked by GR.

 

I fear you are adding unnecessary complexity to a subject which certainly doesn't need it. And I believe DrR will tell you that an embedding dimension is not required by GR.

 

That being said, I believe M theory postulates branes which are embedded in higher dimensional spaces, so we'll just have to wait and see if and how that pans out. And I suppose you can guess my opinion on that.

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Can the fabric of space-time be considered, to be 'embedded', in a higher-dimensional 'hyper-space' ?

 

 

Yes, there is a generalization of the Nash embedding theorem that shows that any Lorentzian manifold can be embedded in a hyper-space of suitably high dimension so as to preserve the metric.

 

But it is rather pointless to do that. You lose the flavor of spacetime as an intrinsic Lorentzian manifold and gain nothing in the process.

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Maybe its just my opinion, and I would certainly appreciate your and others' input on the matter, but have extra dimensions become a 'catch-all' for physics problems lately.

 

Some examples.

 

String theory in the 80s has a problem. It gets rid of the infinite probabilities that previous quantum field gravity theories had provided, but it has negative probabilities, which also make no sense. The 'solution' is to provide 7 extra spatial dimensions for the strings to vibrate in, and the negative probabilities cancel out. We are left with a 10S+1T dimensional space where each point of the three large spatial dimensions contains a compacted 7 dimensional Calabi-Yau manifold ( hope I'm using the terminology properly ). Has this solved the problem ?

 

The Many Worlds interpretation of Quantum Theory where the superposition of states of the wave equation is collapsed by observation or any event which 'fixes' the outcome. In this reality or dimension, one state prevails over the other possible states. But in other realities or dimensions each of all other possible states prevail. Shroedinger's cat is alive in this dimension ( I like cats, I have two ), but dead in another dimension. There are infinite other realities and every time a wave function collapses more are 'created'. Does this solve the interpretation problem ?

 

The time travel problem and closed time-like loops allowed by GR, where a past event is changed to alter a future outcome, such as going back in time to prevent yourself from being born, is interpreted as time branching at that point to create another self-consistant reality or dimension. Extra dimensions are created every time one of these loops is created, not necessarily more spatial dimensions, but the same three spatial dimensions are re-created in a differing time-line, implying at least 2 ( or more ) time dimensions. Has this solved the problem ?

 

It seems like extra dimensions or realities ( I've used a rather loose definition of dimensions ) are invoked whenever we need a rug to sweep our problems under. We can't test for or verify these extra dimensions or realities so its a convenient way to get rid of problems and forget about them without really solving them.

 

If this post derails the OP as it has very little to do with it, please feel free to move it.

 

 

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Maybe its just my opinion, and I would certainly appreciate your and others' input on the matter, but have extra dimensions become a 'catch-all' for physics problems lately.

 

Some examples.

 

String theory in the 80s has a problem. It gets rid of the infinite probabilities that previous quantum field gravity theories had provided, but it has negative probabilities, which also make no sense. The 'solution' is to provide 7 extra spatial dimensions for the strings to vibrate in, and the negative probabilities cancel out. We are left with a 10S+1T dimensional space where each point of the three large spatial dimensions contains a compacted 7 dimensional Calabi-Yau manifold ( hope I'm using the terminology properly ). Has this solved the problem ?

 

No, it has not solved the problem. The extra dimensions are necessary for the consistency of string theory. But string theory has still yet to be clearly defined or to provide any new testable prediction. It remains very preliminary.

 

The Many Worlds interpretation of Quantum Theory where the superposition of states of the wave equation is collapsed by observation or any event which 'fixes' the outcome. In this reality or dimension, one state prevails over the other possible states. But in other realities or dimensions each of all other possible states prevail. Shroedinger's cat is alive in this dimension ( I like cats, I have two ), but dead in another dimension. There are infinite other realities and every time a wave function collapses more are 'created'. Does this solve the interpretation problem ?

 

The many worlds interpretation of Hugh Everett has nothing to do with any extra dimensions. There is no "interpretation problem" just another interpretation of quantum mechanics.

 

 

The time travel problem and closed time-like loops allowed by GR, where a past event is changed to alter a future outcome, such as going back in time to prevent yourself from being born, is interpreted as time branching at that point to create another self-consistant reality or dimension. Extra dimensions are created every time one of these loops is created, not necessarily more spatial dimensions, but the same three spatial dimensions are re-created in a differing time-line, implying at least 2 ( or more ) time dimensions. Has this solved the problem ?

 

What in the world are you talking about ? There is no theory in which "extra dimensions are created every time one of these loops is created" and in fact in general relativity loops are not created, they either exist or they do not. The spacetime manifold contains all of time and all of space and there is nothing that is either "created" or "destroyed".

 

Nor does general relativity either require or admit any extra dimensions, though it might be possible to formulate a theory on a higher dimensional manifold.

 

 

 

It seems like extra dimensions or realities ( I've used a rather loose definition of dimensions ) are invoked whenever we need a rug to sweep our problems under. We can't test for or verify these extra dimensions or realities so its a convenient way to get rid of problems and forget about them without really solving them.

 

You have been reading too much nonsense.

 

The consideration of extra dimensions goes back to Kaluza-Klein and the attempt to incorporate electrodynamics into general relativity. That did not work but it did show that additional spatial dimensions offer some potential, from a mathematical perspective, in attempting to formulate theories that may be testable. It may or may not be an approach that will eventually bear fruit, but don't throw the baby out with the bath water.

 

The issue is not whether or not extra dimensions per se are testable, though they may be, but whether or not a mathematically consistent theory which requires extra dimensions makes new physical predictions that are testable. So far that has not happened, but there is no reason to state a priori that such is impossible.

 

We have no direct physical test to show that atoms exist, but the atomic hypothesis has resulted in any number of physical theories that have shown to be incredibly accurate -- not to mention the entire discipline of chemistry. Still no one has any photograph of an atom, crystal "pictures" showing blobs where we think atoms ought to be notwithstanding.

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You may be right about reading nonsense...

However I believe math is the tool we use to build models and the physics has to guide you in using this tool. Sometimes the tool and model becomes more complicated than the physics its trying to describe. And ok, maybe some of my examples were questionable ( and although not theories have been put forth many times by members of this forum, which is why I used them as examples of unnecessary complexity ), but it certainly applies to string theory, which, although mathematically consistant, has become extremely complicated and as you say, has made no testable predictions.

Edited by MigL
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