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Author Topic: Does curved spacetime require additional dimensions?  (Read 7249 times)

Offline cowlinator

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I always had a bit of difficulty completely grasping the concept of curved space-time. 

To simplify it for myself, I think of a 2-dimentional universe with no time.  Let's say this universe is sphere-shaped.  If I were to plaster this universe onto the surface of a 3-dimentional sphere, it implies that the 3rd dimension must exist for the 2d universe to exist in this shape.

Am I thinking about his in the right manner?  Does this example extrapolate correctly to 4d space-time?  Does our universe require a 5th dimension to be curved?
« Last Edit: 28/09/2014 22:22:22 by evan_au »


 

Offline PmbPhy

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Re: Does curved spacetime require additional dimentions?
« Reply #1 on: 21/09/2014 09:58:12 »
I always had a bit of difficulty completely grasping the concept of curved space-time. 

To simplify it for myself, I think of a 2-dimentional universe with no time.  Let's say this universe is sphere-shaped.  If I were to plaster this universe onto the surface of a 3-dimentional sphere, it implies that the 3rd dimension must exist for the 2d universe to exist in this shape.

Am I thinking about his in the right manner?  Does this example extrapolate correctly to 4d space-time?  Does our universe require a 5th dimension to be curved?
No. Spacetime Curvature acts of the 4 dimensions of spacetime. Three of the dimensions are spatial and one is temporal.
 

Offline yor_on

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Re: Does curved spacetime require additional dimentions?
« Reply #2 on: 21/09/2014 09:59:21 »
Depends on what you mean by a curved universe? As far as I get it the universe as such is more or less 'flat', meaning that there is no limit to it that I know. You should, theoretically, be able to move to where we see the 'earliest light', and from there when looking around be able to see a exact (more or less) replica of what you would see looking around from here. So there is no sphere.
 

Offline JohnDuffield

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Re: Does curved spacetime require additional dimentions?
« Reply #3 on: 21/09/2014 10:22:15 »
I always had a bit of difficulty completely grasping the concept of curved space-time.  To simplify it for myself, I think of a 2-dimentional universe with no time.  Let's say this universe is sphere-shaped.  If I were to plaster this universe onto the surface of a 3-dimentional sphere, it implies that the 3rd dimension must exist for the 2d universe to exist in this shape. Am I thinking about his in the right manner?  Does this example extrapolate correctly to 4d space-time?  Does our universe require a 5th dimension to be curved?
I'm afraid it's a no.

Curved spacetime is easier than you think. Imagine you can place optical clocks al various locations throughout an equatorial slice through the Earth and surrounding space. You know about gravitational time dilation, and that clocks go slower when they're lower. So you know that when you plot all your clock rates, what you get is a plot like this: 


GNUFDL image by Johnstone, see Wikipedia 

That's a depiction of Riemann curvature which relates to curved spacetime. It's important to note that space isn't curved per se. See Baez and note "not the curvature of space, but of spacetime". Curved spacetime isn't curvature of space and curvature of time. It's a curvature in your plot of measurements of motion through space over time. It's a curvature of "the metric", metric being to do with measurement.   
 

Offline PmbPhy

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Re: Does curved spacetime require additional dimentions?
« Reply #4 on: 21/09/2014 11:06:14 »
Quote from: yor_on
Depends on what you mean by a curved universe? As far as I get it the universe as such is more or less 'flat', meaning that there is no limit to it that I know.
You're thinking of space being flat. The spacetime is still curved.
 

Offline cowlinator

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Re: Does curved spacetime require additional dimentions?
« Reply #5 on: 21/09/2014 23:32:42 »
"not the curvature of space, but of spacetime". Curved spacetime isn't curvature of space and curvature of time. It's a curvature in your plot of measurements of motion through space over time.

Ok, that actually makes a lot more sense. 

you know that when you plot all your clock rates, what you get is a plot like this: 


GNUFDL image by Johnstone, see newbielink:http://en.wikipedia.org/wiki/Riemann_curvature_tensor [nonactive] 

So the vertical "dimension" (not a real dimension) seen in this graph is the time dilation?

Quote from: yor_on
Depends on what you mean by a curved universe? As far as I get it the universe as such is more or less 'flat', meaning that there is no limit to it that I know.
You're thinking of space being flat. The spacetime is still curved.

Then what are people referring to when they talk about the shape and/or curvature of the universe?  Are they talking about the curvature of space or of spacetime?
 

Offline JohnDuffield

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Re: Does curved spacetime require additional dimentions?
« Reply #6 on: 22/09/2014 08:34:40 »
So the vertical "dimension" (not a real dimension) seen in this graph is the time dilation?
Yes. Clocks go lower when they're lower. The clock rate relates to gravitational potential. See Wikipedia where you can see a similar depiction.

Then what are people referring to when they talk about the shape and/or curvature of the universe?  Are they talking about the curvature of space or of spacetime?
IMHO there's quite a lot of confusion about that.
 

Offline cowlinator

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Re: Does curved spacetime require additional dimentions?
« Reply #7 on: 27/09/2014 05:27:50 »
Then what are people referring to when they talk about the shape and/or curvature of the universe?  Are they talking about the curvature of space or of spacetime?
IMHO there's quite a lot of confusion about that.
This is why we can't have nice theories!
 

Offline cowlinator

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Re: Does curved spacetime require additional dimentions?
« Reply #8 on: 27/09/2014 21:40:41 »
You were responding to a well-known crackpot when you posted opinion.
He's the only one who answered my question to any degree of detail, and with any degree of clarity.  To me, this implies there is some confusion about it. 
 

Offline Bill S

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Re: Does curved spacetime require additional dimentions?
« Reply #9 on: 28/09/2014 01:20:43 »
Cowlinator, never lose sight of the fact that word meanings in English are determined on the basis of common usage by educated people.  It could be that “crackpot” has evolved to mean someone who takes the trouble to give more than a dogmatic answer to a question.
 

Offline PmbPhy

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Re: Does curved spacetime require additional dimentions?
« Reply #10 on: 28/09/2014 02:12:59 »
You were responding to a well-known crackpot when you posted opinion.
He's the only one who answered my question to any degree of detail, and with any degree of clarity.  To me, this implies there is some confusion about it.
In the reply he used what is often referred to as the rubber sheet analogy. It's real name is embedding diagram. For details please see http://en.wikipedia.org/wiki/Gravity_well#The_rubber-sheet_model

When JD posted the diagram he did above in reply #3 he used what is known as an embedding diagram. That the diagram shows Earth sitting on it tells you that this is an embedding diagram.  Such embedding diagrams represent the relationship between physical distance and proper distance.
« Last Edit: 28/09/2014 22:48:10 by evan_au »
 

Offline cowlinator

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Re: Does curved spacetime require additional dimentions?
« Reply #11 on: 28/09/2014 05:37:21 »
It could be that “crackpot” has evolved to mean someone who takes the trouble to give more than a dogmatic answer to a question.
=D

He's the only one who answered my question to any degree of detail, and with any degree of clarity.  To me, this implies there is some confusion about it.
The number of posts a person makes having what appears to be greater detail cannot logically be used to infer that a person isn't a crackpot.
That's nice.  The only thing I was logically inferring was: the lack of posts, having what appears to be greater detail, that anyone makes, infers that there is some confusion on the subject. 

Quote from: JohnDuffield
So you know that when you plot all your clock rates, what you get is a plot like this
This will mislead you to misinterpret every embedding diagram that you see from now on to be what he claims it means and thus you'd be misusing them too.
So what you're saying is, that because a 3d graph an Earth on it was used to represent certain axes, that means that for every 3D graph with an Earth on it that I ever see, I will assume the same axes. 
What is the basis of your reasoning exactly?
And should I be offended?...
« Last Edit: 28/09/2014 05:39:41 by cowlinator »
 

Offline PmbPhy

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Re: Does curved spacetime require additional dimentions?
« Reply #12 on: 28/09/2014 06:40:04 »
Quote from: cowlinator
That's nice.  The only thing I was logically inferring was: the lack of posts, having what appears to be greater detail, that anyone makes, infers that there is some confusion on the subject. 

In a lot of the cases when I choose not to post on a subject is because adding something would make things worse. For example; when I believe that all that can be said about something such as the concept of time then adding more would be undesirable, not because I'm confused or because there is confusion but because all that can be said has already been said. Anything more is just rambling.

Quote from: cowlinator
So what you're saying is, that because a 3d graph an Earth on it was used to represent certain axes, ...
No. He claimed an embedding diagram is a clock rate diagram, and it isn't. They are different diagrams. If we calculate the embedding diagram then what we get is a plot of a surface which is defined and described chapter 3 of Exploring Black Holes. It's online at  http://www.eftaylor.com/exploringblackholes/Curving140829v2.pdf

As you can see from the derivation you get a surface of the form Z = sqrt(Ar + B)

The derivation of the surfaced that JD is talking about is different. For what follows please see Eq. (3) at - http://home.comcast.net/~peter.m.brown/gr/grav_red_shift.htm

If we let the time lapsed between two events at the spatial position r = (x, y, z) be tr and the time lapsed on a clock located at infinity be tthen

tr = t sqrt(1 - 2GM/r)

then you can easily see that it's is not proportional to Z as JD claims. Do you see my point now?

Quote from: cowlinator
And should I be offended?
I don't know. Should you? I'm at a loss as to why you'd ask and make it so small that anybody not responding would never see it. Why is that? I.e. why did you ask this and who were you asking it to and what about?
« Last Edit: 28/09/2014 09:03:08 by PmbPhy »
 

Offline JohnDuffield

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Re: Does curved spacetime require additional dimentions?
« Reply #13 on: 28/09/2014 14:02:24 »
Apologies. Here's another image, this time showing gravitational potential:



Again, imagine you place light clocks throughout a horizontal slice of space that goes through the Earth's equator. When you plot the clock rates, you get a picture like this. The degree of slope at some location indicates the strength of gravity at that location. The curvature you can see relates to Riemann curvature, which is what curved spacetime is all about.   

See what I said in this thread about the "shape" of the universe.
« Last Edit: 28/09/2014 22:42:50 by evan_au »
 

Offline PmbPhy

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Re: Does curved spacetime require additional dimentions?
« Reply #14 on: 28/09/2014 14:35:13 »
Please look up "gravitational well" which is what the diagram illustrates. To see this please see the diagrams in

http://en.wikipedia.org/wiki/Gravity_well
http://en.wikipedia.org/wiki/Gravitational_potential

and you'll see that the diagrams are identical in every way. However in the first link it explains that
Quote
Both the rigid gravity well and the rubber-sheet model are frequently misidentified as models of general relativity, due to an accidental resemblance to general relativistic embedding diagrams,...
There is an accidental resemblance of gravitational wells and embedding diagrams etc.

This mistake is described here http://en.wikipedia.org/wiki/Gravity_well
Quote
In astrophysics, a gravity well is specifically the gravitational potential field around a massive body. Other types of potential wells include electrical and magnetic potential wells. Physical models of gravity wells are sometimes used to illustrate orbital mechanics. Gravity wells are frequently confused with embedding diagrams used in general relativity theory, but the two concepts are distinctly separate, and not directly related.
....
Gravity wells and general relativity

Both the rigid gravity well and the rubber-sheet model are frequently misidentified as models of general relativity, due to an accidental resemblance to general relativistic embedding diagrams,[citation needed] and perhaps Einstein's employment of gravitational "curvature" bending the path of light, which he described as a prediction of general relativity. In particular, the embedding diagram most commonly found in textbooks (an isometric embedding of a constant-time equatorial slice of the Schwarzschild metric in Euclidean 3-dimensional space) superficially resembles a gravity well.

Embedding diagrams are, however, fundamentally different from gravity wells in a number of ways. Most importantly, an embedding is merely a shape, while a potential plot has a distinguished "downward" direction; thus turning a gravity well "upside down" (by negating the potential) turns the attractive force into a repulsive force, while turning a Schwarzschild embedding upside down (by rotating it) has no effect, since it leaves its intrinsic geometry unchanged. Geodesics following across the Schwarzschild surface would bend toward the central mass like a ball rolling in a gravity well, but for entirely different reasons. There is no analogue of the Schwarzschild embedding for a repulsive field: while such a field can be modeled in general relativity, the spatial geometry cannot be embedded in three dimensions.

The Schwarzschild embedding is commonly drawn with a hyperbolic cross section like the potential well, but in fact it has a parabolic cross section which, unlike the gravity well, does not approach a planar asymptote.

JD's diagram confuses visual appearances rather than mathematical comparison or analysis.
« Last Edit: 28/09/2014 22:41:09 by evan_au »
 

Offline PmbPhy

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Re: Does curved spacetime require additional dimentions?
« Reply #15 on: 28/09/2014 20:42:37 »
Quote from: JohnDuffield
You know about gravitational time dilation, and that clocks go slower when they're lower.
Dear cowlinator,

Contrary to how JD makes it appear the way he phrased it, all clocks run at the same rate in a gravitational field. The only thing that gravity alters in this respect is the local rate at which clock's tick. I.e. when you compare the lack at which two clocks are running will you find a difference. For example; if as measured from a far away observer will you measure all clocks to run at the same rate. That's what gravitational time dilation is.
 

Offline Bill S

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Re: Does curved spacetime require additional dimentions?
« Reply #16 on: 28/09/2014 22:06:40 »
Pete, I have learned quite a lot from you, but I often have to ask you to simplify your explanations.

I’m not trying to take sides in a discussion about which I am unsure of my understanding, I’m just trying to clarify points that I’m not sure about.  What, if anything is wrong with the following?

JD says:

1. You know about gravitational time dilation, and that clocks go slower when they're lower.

2. So you know that when you plot all your clock rates, what you get is a plot like this:  {He then shows a diagram, which may, or may not, be the one he should have chosen, but which does have the right shape to demonstrate what he is explaining.}

3. That's a depiction of Riemann curvature which relates to curved spacetime.

4. Curved spacetime isn't curvature of space and curvature of time. It's a curvature in your plot of measurements of motion through space over time. It's a curvature of "the metric", metric being to do with measurement.   
 

Offline PmbPhy

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Re: Does curved spacetime require additional dimentions?
« Reply #17 on: 29/09/2014 03:06:28 »
Quote from: Bill S link
JD says:

1. You know about gravitational time dilation, and that clocks go slower when they're lower.
There is a specific meaning to this and if not careful one can get into trouble. Suppose we create clocks whose rate at which they tick is set by pulses of light it receives from a distant source. Then the distant source, which we call a "Schwarzschild observer" sends signals at intervals of 1 second to each clock. Since the time between the signals don't change as it passes through the field the clocks dispersed through the field will also run at 1 second between ticks.

It is the comparison between the rates of two clocks in relative position that counts. I explained this in my page at http://home.comcast.net/~peter.m.brown/gr/grav_red_shift.htm  In it I wrote
Quote
What’s wrong with Schild’s argument? First one needs to be careful when interpreting the statement "The frequency of light decreases..." Caution must be exercised when using "the" when discussing relativity. The frequency reckoned but which observer? Which clock is used to reckon this change?
Please read the entire page to understand that snippet. I posted that link in my post "Reply #12". It's a common misconception to claim that the frequency of light changes as it moves through a gravitational field. It's only when comparing the rates of two clocks when compared locally do you measure a change.

The image shown by JD is embedding diagram and does not have the shape that he claims implies does. And since I've challenged him on this he's never made an attempt to show that he's correct and I'm wrong and calculate the exact shape of the surface he's talking about.

Quote from: Bill S link
2. So you know that when you plot all your clock rates, what you get is a plot like this:  {He then shows a diagram, which may, or may not, be the one he should have chosen, but which does have the right shape to demonstrate what he is explaining.}
Why would you think that it has the right shape?

Quote from: Bill S link
3. That's a depiction of Riemann curvature which relates to curved spacetime.
What he posted is an embedding diagram. An embedding diagram is a diagram that depicts only spatial relations and has nothing to do with time. I posted the chapter to exploring black holes so people can read what they're for rather than what JD claims they mean. Embedding diagrams are highly misleading to the untrained. This page explains many of the reasons why its abused
http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/index.html

Please understand this - I wanted to make it 100% clear that the embedding diagram that JD posted is not the diagram he is talking about and although the diagrams might have some similarity in looks it'd be a terrible mistake to think that they're the same diagram hence having the same functional values throughout the diagram. I've made that point as clear as I've been able to in this thread so I can't be accused of being overly picky. The importance is to understand the differences and the diagrams and their relative purposes. Understand?

Quote from: Bill S link
4. Curved spacetime isn't curvature of space and curvature of time. It's a curvature in your plot of measurements of motion through space over time. It's a curvature of "the metric", metric being to do with measurement.
That statement is gobbledygook. The way he wrote it says the following "Spacetime is a plot of measurements of motion ....." and that's not true at all. He's referring to a worldline in spacetime rather than spacetime itself. Spacetime is a manifold in which objects may or may not be moving. E.g. a free object at the origin of a locally inertial frame of reference will never move by definition.

Any more questions? When my forum is up and running and several of you will be invited to join the forum.
« Last Edit: 29/09/2014 12:40:01 by evan_au »
 

Offline JohnDuffield

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Re: Does curved spacetime require additional dimentions?
« Reply #18 on: 29/09/2014 08:39:18 »
Contrary to how JD makes it appear the way he phrased it, all clocks run at the same rate in a gravitational field
This simply isn't true either. See this interview with David Wineland of NIST:

"For example, gravity affects the rate that clocks run. One of the effects of gravity comes from Einstein's theory of general relativity. And one of the consequences of Einstein's theory of general relativity was that clocks, if they're placed near a gravitational mass, say the Earth—will run at a slower rate than if they're removed from the source—say clocks on a satellite. But nowadays the precision of the clocks is such that we have to worry, when we compare clocks, if one clock in one lab is 30 centimeters higher than the clock in the other lab, we can see the difference in the rates they run at."
« Last Edit: 29/09/2014 12:41:54 by evan_au »
 

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Re: Does curved spacetime require additional dimentions?
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