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FAQ article development, feel free to ask questions or make suggestions.

(I'm still working on the Einstein field equation section. Probably keep that portion seperate to minimize length)

This question is amongst one of the most commonly asked questions in relativity. Numerous articles both in pop media and peer reviewed articles refer to terms such as space time fabric, space time curvature. This leads the new learners with a common misconception that space has some mysterious fabric or material like property.

To answer this properly we need to describe a few principles.

A) gravity influences mass

B) energy is a property of particles, or physical configurations such as feilds. Energy does not exist on its own.

C) space is defined as a volume only. That volume contains the standard model particles and feilds. It is not something form of ether. In GR space is mapped in an arbitrary coordinate system. Without the time component the coordinates are in 3d.

D) spacetime is any metric that includes the time component as a vector. This is the 4th dimension, in GR the time component is treated in coordinate form.

E) General relativity is a coordinate system metric. This coordinate system makes use of manifolds. Which is a topological space that is resembles Euclidean space at beach point. For example a Euclidean space (flat space), can undergo a homeomorphism to curved space via relativistic effects such as inertia and mass to an observer.

The rubber sheet example is one such homeomorphism.

http://theory.uwinnipeg.ca/users/gabor/black_holes/slide5.html

Keep in mind the rubber sheet analogy is just that. An analogy, it was never intended to state that space time is a materialistic fabric or ether.

A classical example of a homeomorphism is the coordinate change from Cartesian coordinates (Euclidean flat space) to polar coordinates. (Curved, spherical geometry)

https://www.mathsisfun.com/polar-cartesian-coordinates.html

http://en.m.wikipedia.org/wiki/Manifold

http://en.m.wikipedia.org/wiki/Homeomorphic

Now with those in mind, we find that spacetime curvature is a geometric coordinate relation of how the strength of gravity influences the particles that reside in the volume of space. In short it is a geometric description of how gravity influences particles not the volume of space.

The terms fabric, curvature, sretches are misleading. They are analogies used to explain the change in geometric relations.

2) How is space time created?

The volume of space simply increases, space itself is just volume filled with the standard model particles.

Edited by Mordred
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If space is just volume, how do reconcile 'quantum foam' or are you intending to make a clear distinction between what GR says and QP says?

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Quantum foam is also a mathematical descriptive. When you get into the details. I think it may be best to place that under the QM forum. Then provide a link to each article.

Still deciding on that. Atm the curvature and stress energy tensors is giving me headaches trying to simplify them.

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B) energy is a property of mass, it does not exist on its own.

Energy is a property of physical configurations. This would include fields, massless or not.

C) space is defined as a volume only.

I am not quite sure what is meant by this mathematically. The general notion of a space is probability too loose to be much help, but in general relativity we are dealing with smooth manifolds with the additional structure of a (pseudo)-Riemannian metric. With this metric one can construct a canonical volume and indeed define integrals etc. However, generically I would say that a space (as we mean it here) is a collection of points together with some topology.

In relativity, we can interpret these points as possible locations for 'physical events'; maybe simply the possible locations of a test particle or something.

D) spacetime is any metric that includes the time component as a vector.

Okay, so a space-time is a pesudo-Riemannian manifold. This means that at every point the metric can be brought into the diagonal form (-1,1,1,1) (depending on your conventions). The point is one of the components has a different sign and this signals we have space and time. Note that this does not imply we have some canonical decomposition into space and time, just that it is always possible locally to do this.

E) General relativity is a coordinate system metric.

The solutions to the field equations are metrics on given smooth manifolds. The problem is that the field equations give no direct information on the global topology of the space-time, we can at best search for consistent topologies. There are topological obstructions to the existence of Lorentzian signature metrics.

If space is just volume, how do reconcile 'quantum foam' or are you intending to make a clear distinction between what GR says and QP says?

You would hope that any quantum theory of space-time will in some sensible limits reproduce the classical notions of space and time. So, provided we are not near the scale of quantum gravity, which is the Planck scale (though it is possible that this scale could be much lower), we can safely assume our classical description is good.

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Quantum foam is also a mathematical descriptive. When you get into the details. I think it may be best to place that under the QM forum. Then provide a link to each article.

Still deciding on that. Atm the curvature and stress energy tensors is giving me headaches trying to simplify them.

As very much a novice, I can say this is a biggie and probably very important to get across in some understanble way to those not familiar with them.

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As very much a novice, I can say this is a biggie and probably very important to get across in some understanble way to those not familiar with them.

The idea of parallel transport of vectors and curvature is easy to demonstrate in 2d. We can show this 'Blue Peter' style.

Take a ball or balloon. Mark three distinct points on it and joint them with lines as directly as you possibly can. You will get a 'triangle' but now on the surface of your sphere. Next take a pencil and place it on one of the points. Now move the pencil round the path you have drawn so that it returns to the initial point. You must do this carefully without changing the pencils direction, do not add any rotation by yourself, just let the pencil follow the path.

If you do this carefully, you will see that the pencil does not return to exactly the same configuration as you started. The pencil should be pointing in a different direction. It will be 90 degrees rotated as compared with the starting configuration. This angle is the curvature.

They the same on a flat sheet of paper and also a cylinder. You will then have proved that a cylinder is flat!

Loosley, the pencil represents a vector and moving it round the said loop without changing the vector represents parallel transport. The deficit angle is the curvature.

In higher dimensions the idea is the same, just you have to make this all more mathematical and defined properly. But the basic idea is the same.

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On 5/31/2015 at 1:45 AM, ajb said:

Energy is a property of physical configurations. This would include fields, massless or not.

that statement should have read a property of particles not mass lol oops. Missed that.

On 5/31/2015 at 1:45 AM, ajb said:

I am not quite sure what is meant by this mathematically. The general notion of a space is probability too loose to be much help, but in general relativity we are dealing with smooth manifolds with the additional structure of a (pseudo)-Riemannian metric. With this metric one can construct a canonical volume and indeed define integrals etc. However, generically I would say that a space (as we mean it here) is a collection of points together with some topology.

In relativity, we can interpret these points as possible locations for 'physical events'; maybe simply the possible locations of a test particle or something.

I am working on this aspect, originally I had planned on including the metric and curvature tensor as defined in an arbitrary coordinate system of a point (test particle). The problem I've run into is simplifying the metric for the average reader.

On 5/31/2015 at 1:45 AM, ajb said:

Okay, so a space-time is a pesudo-Riemannian manifold. This means that at every point the metric can be brought into the diagonal form (-1,1,1,1) (depending on your conventions). The point is one of the components has a different sign and this signals we have space and time. Note that this does not imply we have some canonical decomposition into space and time, just that it is always possible locally to do this.

The solutions to the field equations are metrics on given smooth manifolds. The problem is that the field equations give no direct information on the global topology of the space-time, we can at best search for consistent topologies. There are topological obstructions to the existence of Lorentzian signature metrics.

True, again the problem is keeping the article simple yet accurate. I agree more detail on the coordinate aspects of GR is needed for the article, which may be best to apply the Lorentz transformation from two examples from flat and in the Schwartchild metric.

On the quantum foam aspects, it's looking like a link to a separate thread may be best.

On note on the metric section here is what I have thus far and I'm reconsidering how to go about this section.

GR matrix transformations

In General Relativity the metric is seemingly complex. One must understand that GR is a coordinate system. When one describes bodies in motion such as planets and stars the metric of a sphere is useful. However at some point one must use an arbitrary coordinate metric. Recalling that GR has the time component as a coordinate as well. Coordinates in GR take the form (ct,x,y,z) this leads to a 4x4 matrix. For the moment we are ignoring everything but the exact specific real numbers the components of the metric take at a single point. Lets define a point as $x^\alpha$ and our new coordinate as $y^{\mu}$

$g_{\mu\nu}=g_{\alpha\beta}=\frac{dx^{\alpha}}{dy^{\mu}}\frac{dx^{\beta}}{dy^{\nu}}$

What exactly is a matrix. The wiki definition is useful.

"In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns that is treated in certain prescribed ways. The individual items in a matrix are called its elements or entries. "

One example is below. Which is a 4*4 matrix Note the numeric organization.

$A_{m,n} =\begin{pmatrix}a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\\vdots & \vdots & \ddots & \vdots \\a_{m,1} & a_{m,2} & \cdots a_{m,n}\end{pmatrix}$

In GR it is common to replace m and n with $\mu$ and $\nu$ respectively. As one can see $\mu$ denotes the row and $\nu$ denotes the column. Both $\mu$ and $\nu$ are vectors. Matrix transformation examples can be found here

A more detailed 63 page article on matrix mathematics can be studied in this pdf.

Einstein field equation

Metric tensor

In general relativity, the metric tensor below may loosely be thought of as a generalization of the gravitational potential familiar from Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as distance, volume, curvature, angle, future and past.

$dx^2=(dx^0)^2+(dx^1)^2+(dx^3)^2$

$g_{\mu\nu}=\begin{pmatrix}g_{0,0}&g_{0,1}&g_{0,2}&g_{0,3}\\g_{1,0}&g_{1,1}&g_{1,2}&g_{1,3}\\g_{2,0}&g_{2,1}&g_{2,2}&g_{2,3}\\g_{3,0}&g_{3,1}&g_{3,2}&g_{3,3}\end{pmatrix}=\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$

Which corresponds to

$\frac{dx^\alpha}{dy^{\mu}}=\frac{dx^\beta}{dy^{\nu}}=\begin{pmatrix}\frac{dx^0}{dy^0}&\frac{dx^1}{dy^0}&\frac{dx^2}{dy^0}&\frac{dx^3}{dy^0}\\\frac{dx^0}{dy^1}&\frac{dx^1}{dy^1}&\frac{dx^2}{dy^1}&\frac{dx^3}{dy^1}\\\frac{dx^0}{dy^2}&\frac{dx^1}{dy^2}&\frac{dx^2}{dy^2}&\frac{dx^3}{dy^2}\\\frac{dx^0}{dy^3}&\frac{dx^1}{dy^3}&\frac{dx^2}{dy^3}&\frac{dx^3}{dy^3}\end{pmatrix}$

The simplest transform is the Minkowskii metric, Euclidean space or flat space. This is denoted by $\eta[$

Flat space $\mathbb{R}^4$ with Coordinates (t,x,y,z) or alternatively (ct,x,y,z) flat space is done in Cartesian coordinates.

In this metric space time is defined as

$ds^2=-c^2dt^2+dx^2+dy^2+dz^2=\eta_{\mu\nu}dx^{\mu}dx^{\nu}$

$\eta=\begin{pmatrix}-c^2&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$

In an effort to keep this article to a manageable length I will refer to the wiki article on Lorentz transformations and its connection to SR.

A free textbook (open source) can be found here

(For the Schwartzchild Metric I was thinking of using Kruskal Szekeres coordinates.)

Though it may better to stick to the Schwartzchild Metric) and just link other coordinate systems of note.

Edited by Mordred
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I incorporated some of the suggested changes. In the opening post. Please review. Any suggestions welcome including syntax and writing style.

Don't worry I don't bite, I fully expect a site forum FAQ article to undergo numerous adjustments.

Edited by Mordred
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I would write the transformation law for the components of a metric as

$g_{\mu' \nu'} = \frac{\partial x^{\rho}}{\partial x^{\mu'}}\frac{\partial x^{\epsilon}}{\partial x^{\nu'}}g_{\epsilon \rho}$,

for changes of coordinates $x^{\mu'} = x^{\mu'}(x)$ (with the standard abuses of notation).

I don't quite follow what you have written, but I expect this is just a notational thing.

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Lol if you didn't understand or follow that article portion. Regardless of notation or otherwise. Then that portion definetely needs a major revamp and rethink.

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• 11 months later...

Great info. Why is the transformation of cartesian to polar coordinates so important to the eventual curvature of space-time? I see that polar coordinates seem to represent a subjective reality radiating out from a singularity at the center, which is presumably, the observer and experiencer, and the cartesian coordinates seem to represent an objective view, but why does switching between them become such a key in figuring the curvature of space?

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In short, the physics should not critically depend on the choice of local coordinates.

I don't quite follow you question, but one must be comfortable changing coordinates in geometry and physics.

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It seems that either the polar or the cartesian coordinate systems could be used to accurately map out curves and positions in space. Why bother switching between them?

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It seems that either the polar or the cartesian coordinate systems could be used to accurately map out curves and positions in space. Why bother switching between them?

If a system naturally has a spherical symmetry, then it mathematically the solutions to the problems at hand may be neater and easier to obtain. But you are essentially right, any coordinate system could be used.

In general relativity certain choices of coordinate systems may be easier to physically interpret than other choices. Basically all our interpretations come down to comparing our curved space-times with what we know on flat space-time. Picking the 'right' coordinates may make this clearer.

Edited by ajb
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The universe consists of energy, that comes in the form of dust ans gas. The coordinates are constantly changing, and you can not make calculations without a flexible system.

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The coordinates are constantly changing, and you can not make calculations without a flexible system.

I do not understand your statement.

Usually one can only pick local coordinates and so it is essential that we know how to change between coordinate systems. But I am not sure what you mean by coordinates constantly changing.

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In post #7, under the metric tensor paragraph, it should read $g_{\mu \nu}$, and not $G_{\mu \nu}$, to avoid confusion with the Einstein tensor. Just a small thing though ##### Share on other sites

Thanks for the catch, gotta love auto corrects. Correction applied.

Edited by Mordred
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• 3 weeks later...

Hi, noob here and new member at the same time.

My take on it.

Space must be made of energy)) Energy is somewhat can be defined as ability to produce work against resistant medium (environment).

what seems to be a present 'structure' or 'made of' might have been expansion epoch where 'string stage' gave birth to gluons and what not, and no longer present. There I think should be elastic resistance medium that energy is penetrating at its highest concentration points.

))

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• 3 months later...

I have the same beef as some others here in the first posts. You may define space as volume only, but the claim that it is "not some kind of ether", is controversial at best.

It directly contradicts Einstein's interpretation of GR. He argued that necessarily space is not physically empty. Of course, you could say that space is not some kind of ether, but instead contains some kind of ether; however most people would probably consider that a word game. Even more, "space" as a locally defined volume depends on the physical properties of that volume.

In a nutshell: " "empty space" in its physical relation is neither homogeneous nor isotropic [..]

The ether of the general theory of relativity is a medium which is itself devoid of all mechanical and kinematical qualities, but helps to determine mechanical (and electromagnetic) events. "

Of course, it's easier to spot wrong or controversial formulations than it is to give right or uncontroversial ones. You could simply leave out the phrases "It is not something form of ether" and "not the volume of space", but that would be a little poor in view of the title. So, you could maybe state that space can be considered as a map (or part of a map, in view of space-time) that relates to a physical reality, some effects of which we describe with the map. Just my 2 cts. Another issue is with point 2: according to QFT, if I'm not mistaken, space is filled with fields. Probably it's better not to include "space time creation" in the topic "what is space made of".

feilds ->fields

Which is a topological space that is resembles Euclidean space at beach point.

-> Which is a topological space that resembles Euclidean space at each point. (right?)

Edited by Tim88
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I believe the statement" space is filled with the standard model particles and fields" is a better statement than space being some form of ether is more accurate.

It is those fields and SM particles that determines the topography.

The point being space itself does not have its own unique particles, which would be required for an ether. Unless you accept the graviton neither does gravity...

Even twistor theory doesn't state space itself has its own particles. Though I had to confirm that with a PH.d that specialized in string/twistor theory.

The metric tensor is determined via SM particle distribution.

Thanks for the spelling corrections.

I'm curious though why you would post a 1922 translation. The details of that paper is outdated by later research.

Edited by Mordred
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I believe the statement" space is filled with the standard model particles and fields" is a better statement than space being some form of ether is more accurate.

[..]

I'm curious though why you would post a 1922 translation. The details of that paper is outdated by later research.

That is better, however as I understand from Neumaier's book on QFT, it's inaccurate to put "the standard model particles and fields"; particles are assumed to be phenomena caused by fields. Maybe if you want to go beyond pure GR, you have to choose your favourite quantum theory! And if I'm not mistaken, even in absence of such particles or fields, the speed of light and the lengths of rulers are supposedly determined in GR.

Further, "outdated" could merely refer to fashion; that does not suffice for making pertinent claims as if giving factual statements. So, now I am curious what of Einstein's paper has convincingly been disproved by later research. BTW, a similar paper stems from 1924 but it's not yet available on Wikisource, so for onlookers I chose the more accessible version. Similarly I often cite Einstein's 1905 paper for SR as it's in some points clearer and more precise than a number of later papers by other authors.

Edited by Tim88
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Good point on particles and fields. I was considering a rewrite on that post. Unpinning this thread and replacing the article. It had initially hoped for a more collaborative on its initial writing.

A couple of details I wish to add is a decent example of many particle distribution and how it correlates to the metric tensor and geodesics.

The problem isn't that I can't derive the necessary equations. It's finding a series of derivitaves that can be readily followed.

Specifically thinking the Newtonian limit where $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$

I'm certainly open to other suggestions. However I feel this may be a good way of showing the field aspects of mass only.

In a way I've been posting numerous derivitaves along these lines in Speculations and relativity forum to test how well they are understood by others.

Edited by Mordred
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• 3 months later...

If the variables in a system were relative to each other, That relativity could turn into a dimension like space. The differences in the relativity would be seen as "distance" by the creatures living on that plane.

Hope that's understandable.

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If the variables in a system were relative to each other, That relativity could turn into a dimension like space. The differences in the relativity would be seen as "distance" by the creatures living on that plane.

Hope that's understandable.

Not exactly but I think you are referring to what is known as intrinsic geometry.

Perhaps if you would like to explain in greater detail?

This extract from

Elementary Geometry : Roe : Oxford University Press

might help. Read paragraph 12.1 in particular. Edited by studiot

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