What is Space? A Theory Exploration

[cAm]

First off, sorry if this is the wrong forum, obviously I'm new here, so not sure quite how things go

Anyway, I've been thinking a bit, and wondering exactly What space is, or what the current theory on it is. From what i have thought about myself, it seems that space would have to have a certain density because it has to have volume (doesn't it?), so also, would its density have to be constant? Would it have any relationship to fluids? Most stuff i read doesn't have that much to say about 'space' but it IS there, so it has to be dealt with. From what I've heard about string theory (i haven't read much, so sorry if I'm wrong on this) strings might be considered the building block of Everything, but if that is so, what is BETWEEN the strings? There has to be something between them, because if there was Nothing between things, then they wouldn't be seperated...

thats all i have for now, please, destroy this theory, and replace it w/ a better one :tongue2:

and sorry if this isn't too comprehensible, my brain is shot, just finished phys c2 test and digi elec tests :tongue2:

 

[Rob Woodside]

Anyway, I've been thinking a bit, and wondering exactly What space is, or what the current theory on it is. From what i have thought about myself, it seems that space would have to have a certain density because it has to have volume (doesn't it?), so also, would its density have to be constant? Would it have any relationship to fluids? Most stuff i read doesn't have that much to say about 'space' but it IS there, so it has to be dealt with. From what I've heard about string theory (i haven't read much, so sorry if I'm wrong on this) strings might be considered the building block of Everything, but if that is so, what is BETWEEN the strings? There has to be something between them, because if there was Nothing between things, then they wouldn't be seperated...

Reply:
The short answer is no one knows. People have been dealing with "space" since they were throwing rocks at small animals for food and wondering at the night sky. It wasn't until the Sumerians and Egyptians, concerned with land ownership, started tabulating the sides of right angle triangles that the notion of space began to appear. The sumerians had a different unit for vertical distance than horizontal distance to account for the fact that things fall down and not sideways. The Greeks axiomatized the early results into Euclidean geometry that we now call flat space. This was the first axiomatic theory and it gives a geometry that is different from our visual space. Visually, railroad tracks bend and join at the horizon. If you want to build things you can't just trust your eyes! Some 1500 years later Newton built a theory of motion of physical bodies on an arena of flat space. This space was immutable and unaffected by the presence of matter. In the nineteenth century mathematicians realized there were alternative geometries that were logically independant from Euclidean geometry. Gauss and Riemann measured the angles in a triangle made by three mountain tops and allowing for Earth's curvature found the angles to sum to 180 degrees, just what Euclid would have suspected. Early last century Einstein realized that the arena for Newtonian mechanics was different than the arena for electromagnetism. To resolve the difference Einstein had to weld space and time together to make space-time and the theory of special relativity. Then thinking deeply about gravitation, Einstein came up with General Relativity. In this theory the distance between any two points depends slightly on the presence of nearby matter. Although one talks of relativity "theory", both special and general are experimentally well justified. More recent ideas of a discrete space-time and spin networks etc do not share that luxury.

So what is space? Is it a real thing or is it a conceptual map that allows us to get about? Is it a material substance with odd properties or a mental construct that lives in our heads and mirrors relations between physical objects? Mathematicians have the same problem with numbers. I would argue that reality consists of physical objects AND the relations between them. People who talk of the "fabric of space-time" are telling you space is a material substance. This raises the possibility of cutting and pasting it to make wormholes like trouser legs. Sadly I doubt this. My personal belief is that space is purely relational.

 

[pmb_phy]

What space is,..

The container of objects, i.e. That which contains all objects.

That's the short version. There is some physics literature on this. Two noted texts are Concepts of Space, by Max Jammer and Space & Time, Hans Reichenbach.

Einstein wrote on this some in his relativity texts. He also spoke a bit on it in the foreword of Jammer's text referenced above. Basically I think his definition was "That which contains all objects" but that summary probably doesn't do it justice.

See also http://www.wfu.edu/~brehme/space.htm

Pete

 

[marcus]

My personal belief is that space is purely relational.

you are in good company
a long history of this idea
from aristotle to descartes, including Leibnitz

space does not have a separate existence, it is not an absolute thing that exists in and of itself
but consists of the relations between things, between events

next to, between this and that, where that happened

In Rovelli book Quantum Gravity (last year's draft is downloadable free fromhis site just google the name "rovelli") he has a discussion of
the history of the idea of space, and different ideas of space (and time)
in physics today.

Particle theorists, quantum field theorists, tend to treat space as an absolute-----some kind of space exists on its own and things happen in it.
Relativists (including quantum gravitists) tend to treat space as relational.

Newton broke with the tradition of relational space and invented an absolute space and time (Rovelli has some interesting quotes, Newton had some misgivings about what he was doing).
Flat Minkowski space is the descendant of Newtonian absolute space.

In 1915 Einstein went back to a relational approach and realized the flat space of his 1905 theory as one possible solution of the GR equation, a solution where there is no matter to curve the space.

In GR points have no physical reality-----there are only things, events, relations between them----rovelli has some interesting quotes from Einstein about this too.

It is a fascinating subject.

I suppose we will understand space better when we understand why matter bends it.

there is a sense in which space is nothing other than the gravitational field
(and the gravitational field can be defined in such a way that it does not require any absolute space in which to be defined)

 

[pmb_phy]

space does not have a separate existence, it is not an absolute thing that exists in and of itself but consists of the relations between things, between events ...next to, between this and that, where that happened

This is related to Mach's Principle and the notion that in the absence of matter there is an absence of space (no metric). It also is related to the specific interpretation of Mach's Principle. I've seen some claim that space exists even in the absence of matter. But I've never agreed with that interpretation.

...there is a sense in which space is nothing other than the gravitational field (and the gravitational field can be defined in such a way that it does not require any absolute space in which to be defined)

How? Please state this definition.

Pete

 

[Rob Woodside]

Quote from Marcus:
...there is a sense in which space is nothing other than the gravitational field (and the gravitational field can be defined in such a way that it does not require any absolute space in which to be defined)

Quote from Pete:
How? Please state this definition.

Reply: I think it more of a perspective than a definition. If you do not see this you are only considering a physical field living in material spacetime. What Marcus is meaning by absolute space is just such a substantial space that exists as a thing in itself. This absolute space is essentially independent from other kinds of matter, but possibly influenced by them. However, spacetime may be more subtle than a physical object or material substance. Reality may consist of physical objects and the patterns they follow or relations they have. Thus the electromagnetic field, current densities and vector potential (up to a gauge transformation) are physical objects and Maxwell's equations are just as real, but not physical objects. They are the real patterns and relations that the physical objects enjoy. Taking fields as physical objects one can imagine some special fields that were non vanishing everywhere and every when. Such physical objects could be the metric (up to gauge transformations) or curvature, then spacetime manifolds are just some of the real relations or patterns enjoyed by the curvature or metric.

 

[QuantumTheory]

I believe space is nothing more than a fabric of time itself. It is a void, where there is matter, dark matter, and anti-particles. We will never figure out EVERYTHING about the universe; there will always be questions.

Everything in this world, on our earth, and in the universe as a whole has a relationship to one another.

For example, to understand why plantets orbit, why black holes work, etc, and to understand space as "a fabric", do this little experiment:

Take a light fabric of some type, and put a somewhat heavy ball in the middle. Then let one person hold onto the fabric, while you put the ball in the middle. Then, take a smaller ball, at least 1/2 size of the larger one, and you'll see that the lighter ball will spin around the heavier one, around and around, until it finally meets it.

Space is the sheet, or fabric, and the ball is any massive object in space. Thus, space and time.

Another fun way to try a fun experiment to have simulate black holes, is to do this:

Take a cup of coffee. Put a spoon in it, stir the coffee ALOT with the spoon. You'll notice the motion from the spoon is direcetd into the coffee, making it whirl around in circles. Then take a small object, like a pea, and put it in the coffee. The pea will whirl around, and around, and around, until it gets to the center, and then dip down into the middle of the "black hole".

In this case, the swirling coffee is the black hole, while the pea is some lighter object ( star, etc) .

 

[Rob Woodside]

I believe space is nothing more than a fabric of time itself. It is a void...

snip

...understand space as "a fabric", ...

snip

...Space is the sheet, or fabric...

snip

.

Reply:

So Quantum Theory thinks that space is merely a fabric of time, a void, the sheet, or fabric. No wonder Einstein thought that Quantum Theory was confused.

When I stir a coffee cup with something floating on the surface, I find that it settles on the rim and not at the centre when the rotation stops. Similarly something like tea leaves that rest on the bottom collect at the centre and not the rim as the rotation stops. Can Quantum Theory explain this contradiction?

 

[pmb_phy]

What Marcus is meaning by absolute space is just such a substantial space that exists as a thing in itself. This absolute space is essentially independent from other kinds of matter, but possibly influenced by them.

This doesn't make sense to me. It doesn't define space and it doesn't yield to a definition that I saw here of "space". Whether this thing exists will depend on what it was that Marcus was speaking of. Its really to vauge for me to see what you and Marcus are talking about.

Thus the electromagnetic field, current densities and vector potential (up to a gauge transformation) are physical objects and Maxwell's equations are just as real, but not physical objects.

This I don't understand, i.e. why you call them physical objects. They describe physical objects. They are not physical objects in themelves. To me that'd be like calling velocity a physical object.

Pete

 

[RawThinkTank]

If had space made of something then it should have been of smallest particles because otherwise nothing could have moved through it so easily. BUT if so then what is inbetween those particles ? So what do U think, Or Can U ?

 

[Gamish]

what if space is actually made of something, such as dark matter? If space can be thickened and thinned, according to SR and GR, then it must have a fabric. This is what I say to it all.

"Matter, energy, Space, Time, Us, and God, we all exist, but we will never understand how or why, until it is revealed to us. certantly not in this life. We see the effects of the entities of the universe, but we don't see how it could have been. God does. The fact is that in the beginning there was something, and if there was nothing, then nothing that exist could have come into existence that exist, and if something did actual come from nothing, then that nothing was really something in the first place. But what is this something, a being who created all, or just random energy or matter?"

 

[Rob Woodside]

This doesn't make sense to me. It doesn't define space and it doesn't yield to a definition that I saw here of "space". Whether this thing exists will depend on what it was that Marcus was speaking of. Its really to vauge for me to see what you and Marcus are talking about.
This I don't understand, i.e. why you call them physical objects. They describe physical objects. They are not physical objects in themelves. To me that'd be like calling velocity a physical object.

Pete

Sure, velocity is a property of physical object. It is a relation that an object has relative to another object. I would claim that it is just as real as the object, but as you point out, it is very different You seem to say that an electromagnetic field is a property of charges and currents and these in turn are properties of physical objects. If so, you are saying reality consists of physical objects and their relations. Presumably we both agree that the relations are just as real, though different from physical objects. We differ on just which are the relations and which are the objects. For me the electromagnetic field, vector potential (up to a gauge transformation) and current densities are real physical objects which can be singled out by their electromagnetic relations (Maxwell Equations) and may have other properties as well, such as energy tensors etc. The fact that we can't agree on whether or not the electromagnetic field is a physical object or a relation is exactly the same situation as with space. Is it a physical object like a fabric or merely a set of relations?

Gamish, Don't bring God into this. It is confused enough already. As Laplace said to Napoleon, when questioned about not even mentioning God in the Mechanique Celeste, "Sire, I had no need for the hypothesis."

 

[donjennix]

you are in good company
a long history of this idea from aristotle to descartes, including Leibnitz
space does not have a separate existence, it is not an absolute thing that exists in and of itself but consists of the relations between things, between events
next to, between this and that, where that happened

This is really the essence of the answer and I have little to add; I would observe that the current cosmological models (by current I mean circa 2000, 2001 popularizations) seem to have "space" expanding faster than c (Lawrence Krauss, "Quintescence") -- an interpretation that was confirmed by a recent PhD in physics from Cal Tech. Exactly what that means is something I tend to ignore.

 

[yogi]

I will take issue with most of the above. While space does not appear to be made of anything that we can identify with matter or material particles - it does have important properties. It has a capacitance per unit length and and inductance per unit length (like a tranmission line) Together these properties determine the velocity of radio waves, and its characteristic impedance - antenna designers know they must match the characteristic impedance 377 ohms to the impedance of antennas in order to get maximum power transfer. There have always been two opposing views of space - like those of Libnetz and berkely that space was only a framework reference - a sideless box as Berkely called it - but on the other side there is Einstein, Dirac and Newton. The most convincing evidence to me is in the concept of inertia - accelerate a mass wrt space and you get an instantaneous reactionary force.

 

[Janitor]

...My personal belief is that space is purely relational.

Some time back I was looking at a website that discussed this idea. If I can find it again, I will link to it. The author claimed that as Einstein developed General Relativity, he hoped it would be purely relational, in line with Mach's thoughts. But when the dust settled, Einstein realized that his theory was not purely relational. The author of the website was doubtful that any truly 'purely' relational scheme could ever describe our actual universe. He did mention in passing that a couple of people had written articles (books?) that pushed relationalism as far as it could go. If I had to try to come up with names, it may have been Barbour and Bertotti. Anybody heard of them?

 

[Loren Booda]

Relative to us are an infinity of spaces that follow their own physics - some accessible to us, and others not.

 

[force5]

My view

Right or wrong, this is the direction my research has taken me;

Space is an extention of its source. The source is like one side of the coin and the space is the other side. When energy radiates away from its source, the total mass of the object is constantly being transformed into space. This will result in a change in either the volume or the density of the space in question. If this is the case, then, our solar system would extend out about 5 billion LY's in all directions and have some gravitational affect on any mass within that radius. Anyway, I havn't found a better explanation.

Just my thoughts...John A.

 

[Rob Woodside]

I will take issue with most of the above. While space does not appear to be made of anything that we can identify with matter or material particles - it does have important properties. It has a capacitance per unit length and and inductance per unit length (like a tranmission line) Together these properties determine the velocity of radio waves, and its characteristic impedance - antenna designers know they must match the characteristic impedance 377 ohms to the impedance of antennas in order to get maximum power transfer. There have always been two opposing views of space - like those of Libnetz and berkely that space was only a framework reference - a sideless box as Berkely called it - but on the other side there is Einstein, Dirac and Newton. The most convincing evidence to me is in the concept of inertia - accelerate a mass wrt space and you get an instantaneous reactionary force.

Aristotle thought that space was relational and a space devoid of objects a nonsense. "Nature abhor's a vacuum"

In 1600 Giordano Bruno was burnt at the stake for suggesting that space was infinite. (A satisfying, but unethical method of solving academic disputes)

Newton physically interpreted Descartes analytic geometry as inertial frames. He was aware of Galileo's principle of relativity, but took the frame of the fixed stars as an absolute space. This was the sensorium of God and Newton populated it with aether, a remarkably penetrable substance.

Maxwell, as Yogi says, saw space as not just geometry, but with the properties of a dielectric. But was that space or the aether?

As Laplace dealt with the concept of God, so Einstein dealt with the concept of aether. In support of Mach's ideas, Einstein saw an object "kicking" space by changing the nearby curvature with its mass and space "kicking back" by endowing the object with mass. I have never understood this, but it is treating space as an object or material substance. Certainly one way of spotting a physical object is: if you kick it, it kicks back.

Paradoxically quantum theory treats the vacuum as the plenum. Is it space that is frothing or is it the vacuum fields?

Certainly the understanding of a vacuum will be pivotal in unifying Quantum and Relativistic physics.

 

[Rob Woodside]

Relative to us are an infinity of spaces that follow their own physics - some accessible to us, and others not.

If they are not accessible, how do you know they are there?

 

[Loren Booda]

If they are not accessible, how do you know they are there?

By scientific faith that measurable physics will eventually discover more vectors of communication than just the photon - e. g., the Higgs boson.

Moreover, by statistical argument, an infinitude of spaces exhibits limitless configurations (whether we can observe them or not), assuming a boundless cosmology.

 

[Chronos]

Reverse engineering answer: space is the absence of matter

 

[mijoon]

Some time back I was looking at a website that discussed this idea. If I can find it again, I will link to it. The author claimed that as Einstein developed General Relativity, he hoped it would be purely relational, in line with Mach's thoughts. But when the dust settled, Einstein realized that his theory was not purely relational. The author of the website was doubtful that any truly 'purely' relational scheme could ever describe our actual universe. He did mention in passing that a couple of people had written articles (books?) that pushed relationalism as far as it could go. If I had to try to come up with names, it may have been Barbour and Bertotti. Anybody heard of them?

Perhaps this is what you're thinking of...

"Indeed this was Einstein's eventual answer to Mach's critique of pre-relativity physics. Mach had complained that it was unacceptable for our theories to contain elements (such as spacetime) that act on (i.e., have an effect on) other things, but that are not acted upon by other things. Mach, and the other relationalists before him, naturally expected this to be resolved by eliminating spacetime, i.e., by denying that an entity called "spacetime" acts in any physical way. To Mach's surprise (and unhappiness), the theory of relativity actually did just the opposite - it satisfied Mach's criticism by instead making spacetime a full-fledged element of theory, acted upon by other objects. By so doing, Einstein believed he had responded to Mach's critique, but of course Mach hated it, and said so. Early in his career, Einstein was sympathetic to the idea of relationism, and entertained hopes of banishing absolute space from physics but, like Newton before him, he was forced to abandon this hope in order to produce a theory that satisfactorily represents our observations."


Quoted from this site.

 

[Janitor]

Perhaps this is what you're thinking of...

That is precisely the one!

It also says, "Unfortunately, no completely successful relational theory of motion has ever been devised (although there have been some interesting attempts, cf., Barbour and Bertotti)."

 

[Rob Woodside]

Perhaps this is what you're thinking of...

"Indeed this was Einstein's eventual answer to Mach's critique of pre-relativity physics. Mach had complained that it was unacceptable for our theories to contain elements (such as spacetime) that act on (i.e., have an effect on) other things, but that are not acted upon by other things. Mach, and the other relationalists before him, naturally expected this to be resolved by eliminating spacetime, i.e., by denying that an entity called "spacetime" acts in any physical way. To Mach's surprise (and unhappiness), the theory of relativity actually did just the opposite - it satisfied Mach's criticism by instead making spacetime a full-fledged element of theory, acted upon by other objects. By so doing, Einstein believed he had responded to Mach's critique, but of course Mach hated it, and said so. Early in his career, Einstein was sympathetic to the idea of relationism, and entertained hopes of banishing absolute space from physics but, like Newton before him, he was forced to abandon this hope in order to produce a theory that satisfactorily represents our observations."


Quoted from this site.

This is exactly the point I do not understand. Matter endows space with curvature and space endows matter with inertia. This is not action/reaction force pairs as Yogi suggested. Those are better understood as matter or field interaction and appear independently of the space spanned by the fields. I'm happy with matter producing curvature, but I fail entirely to see space endowing matter with mass. If there was anything to this, one ought to be able to produce mass spectra. I don't know of ANY significant or measurable result that has been deduced from "Space endows matter with inertia" It just seems part of the folklore of relativity.

 

[pmb_phy]

This is exactly the point I do not understand. Matter endows space with curvature and space endows matter with inertia. This is not action/reaction force pairs as Yogi suggested. Those are better understood as matter or field interaction and appear independently of the space spanned by the fields. I'm happy with matter producing curvature, but I fail entirely to see space endowing matter with mass. If there was anything to this, one ought to be able to produce mass spectra. I don't know of ANY significant or measurable result that has been deduced from "Space endows matter with inertia" It just seems part of the folklore of relativity.

This hinges on what one means by "inertia" or "mass". Depending on the definition its possible for the mass of an object to be altered by the presence of matter. This was first introduced by Mach and later elaborated on by Einstein, e.g. see The Meaning of Relativity, Albert Einstein. Einstein uses the definition of inertial as (for a spacetime which is time orthogonal)

$$m = \gamma(v = 0, \Phi)$$

where $\Phi$ is the gravitational potential. There is an interesting related article in Am. J. Phys. on this subject, i.e.

Specific Physical Consequences of Mach's Principle", J. David Nightingale, Am. J. Phys. 45, 376-379 (1977)

Peacock also touches on this point in his text Cosmological Physics. Something about gravitational radiation contributing to the inertial of matter. That part of his text is online at
http://assets.cambridge.org/0521422701/sample/0521422701WS.pdf

See the section labled "Inertial Frames and Mach's Principle".

Pete

 

[yogi]

Here is a quote from Einstein's Leyton address in 1920 which sort of sums up his view of the subject:

“...to deny the ether is ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics to not harmonize with this view. For the mechanical behavior of a corporal system hovering freely in empty space not only depends upon relative positions (distances) and relative velocities, but also on its state of rotation, which physically may be taken as a characteristic not appertaining to the system itself. In order to be able to look upon the rotation of the system, at least formally, as something real, Newton objectivises space. Since he classes his absolute space together with real things, for him rotation relative to absolute space is also something real. Newton might no less well have called his absolute space “Ether”; what is essential is merely that beside observable objects, another thing, which is not perceptible, must be looked upon as real, to enable acceleration or rotation to be looked upon as something real."

"It is true that Mach tried to avoid having to accept as real something which is not observable by endeavoring to substitute in mechanics a mean acceleration with reference to the totality of the masses of the universe in place of an acceleration with reference to absolute space. But inertial resistance opposed to relative acceleration of distant masses presupposes action at a distance; and as the modern physicist does not believe that he may accept this action at a distance, he comes back once more, if he follows Mack, to the ether, which has to serve as medium for the effects of inertia. But this conception of the ether to which we are lead by Mack’s way of thinking differs essentially from the ether conceived by Newton, by Fresnel and by Lorentz. Mack’s ether not only conditions the behavior of inert masses, but is also conditioned in its state by them."

 

[yogi]

Space appears to obey laws that are applicable to fluid dynamics - but unlike a fluid under pressure it appears more like a fluid under tension - something we don't experience with real fluids. The unseen energy required for critical density may exist in the form of tension (stress) - Cosmological Expansion would increases the tension and consequently the stress energy. Alan Guth once ruminated that inflation may be an ongoing phenomena - in which case it is plausible that stress energy is continually converted from negative potential energy at the proper rate to maintain the appearance of critical density at any epoch.

 

[metech00]

Is Space Distance?

I've been struggling with the concept of space as well. I've read the posts here so excuse me if I reiterate some things that have been address but this is the state of confusion I'm in:

Baseline: The big bang is not an explosion but an expansion meaning that space itself is being created. As opposed to an explosion that has a wave front, the big bang happened and is happening everywhere. If you and I are sitting several million light years away, perfectly still with no delta V whatsoever, the distance between us would be increasing but we are not moving. This is the same in all directions; things above us are growing in distance, things on either side, in front, in back and below.

What then, is this 'space'. What exactly is appearing? It's not inert in the sense that things like gravity, light, magnetism, are all effected by distance so this new 'space' impacts things, it has an effect so it must have properties and attributes. What is it and where does it come from?

At the risk of sounding extremely ignorant I would dub this new space as "Distance" for the purposes of discussion. In other words, new Distance is appearing all the time. Distance factors into a lot of equations; why not give it new attributes, properties, and importance.

Since I'm on a -- let's show the world exactly how ignorant I am -- a further question on space and expansion (of course, this assumes that my base-lining in the second paragraph is a true representation of space expansion). Is there a differing rate to the expansion beyond the obvious (my atoms aren't separating etcetera). At some point this expansion is measurable and thus it begins somewhere, what is the demilitarized zone at which this expansion can be measured? Within the solar system, between solar systems? Between Galaxies? In Tuscaloosa? (sorry, it's the Monty Python in me..*sigh* )

 

[Dragongod]

If Space Is Actually All Matter Then Wouldn't That Mean The Universe Is Opaque (completely Solid). ?

 

[Loren Booda]

The microwave background marks a distance beyond which is a space of electrons and protons primarily opaque to photons.

 

[Dragongod]

Loren Booda is that in response to me cause if it is i don't understand what you mean. also when i say matter i don't just mean electrons and protons. I mean all matter i.e. including photons and whateva else.

 

[Loren Booda]

Just an example of opacity.

 

[Garth]

"What is Space?" That which 'G_d' uses to prevent everything happening all in the same place.

"What is Time?" That which 'G_d' uses to prevent everything happening all at once.

"What is 'G_d'? That which is your answer to Stephen Hawking's question "What breathed fire into the equations so that there was a universe for them to describe?"

Garth

N.B. Such a response is not as frivolous as it might first appear. In relating the existence of time and space to 'everything' - the events happening within space-time - it is a relational definition, in the Machian sense.

 

[juju]

Space is a field potential matix underlying all of space time.

It is a dynamic scaler/vector/spin potenial field.

It has its own geometries and symmetries at different fractal levels.

juju

 

[sd01g]

Reverse engineering answer: space is the absence of matter

I like your definition of space. Question: How does energy fit in the definition? Also, what is the difference between space and the 'fabric of space?' Any thoughtful response appreciated.