Is the universe a hologram?

April 27, 2015
Is our universe a hologram? Credit: Vienna University of Technology

Describing the universe requires fewer dimensions than we might think. New calculations show that this may not just be a mathematical trick, but a fundamental feature of space itself.

At first glance, there is not the slightest doubt: to us, the universe looks three dimensional. But one of the most fruitful theories of in the last two decades is challenging this assumption. The "" asserts that a mathematical description of the universe actually requires one fewer dimension than it seems. What we perceive as three dimensional may just be the image of two dimensional processes on a huge cosmic horizon.

Up until now, this principle has only been studied in exotic spaces with negative curvature. This is interesting from a theoretical point of view, but such spaces are quite different from the space in our own universe. Results obtained by scientists at TU Wien (Vienna) now suggest that the holographic principle even holds in a flat spacetime.

The Holographic Principle

Everybody knows holograms from credit cards or banknotes. They are two dimensional, but to us they appear three dimensional. Our universe could behave quite similarly: "In 1997, the physicist Juan Maldacena proposed the idea that there is a correspondence between gravitational theories in curved anti-de-sitter spaces on the one hand and quantum field theories in spaces with one fewer dimension on the other", says Daniel Grumiller (TU Wien).

Gravitational phenomena are described in a theory with three spatial dimensions, the behaviour of quantum particles is calculated in a theory with just two spatial dimensions - and the results of both calculations can be mapped onto each other. Such a correspondence is quite surprising. It is like finding out that equations from an astronomy textbook can also be used to repair a CD-player. But this method has proven to be very successful. More than ten thousand scientific papers about Maldacena's "AdS-CFT-correspondence" have been published to date.

Correspondence Even in Flat Spaces

For theoretical physics, this is extremely important, but it does not seem to have much to do with our own universe. Apparently, we do not live in such an anti-de-sitter-space. These spaces have quite peculiar properties. They are negatively curved, any object thrown away on a straight line will eventually return. "Our universe, in contrast, is quite flat - and on astronomic distances, it has positive curvature", says Daniel Grumiller.

However, Grumiller has suspected for quite some time that a correspondence principle could also hold true for our real universe. To test this hypothesis, gravitational theories have to be constructed, which do not require exotic anti-de-sitter spaces, but live in a flat space. For three years, he and his team at TU Wien (Vienna) have been working on that, in cooperation with the University of Edinburgh, Harvard, IISER Pune, the MIT and the University of Kyoto. Now Grumiller and colleagues from India and Japan have published an article in the journal Physical Review Letters, confirming the validity of the correspondence principle in a flat universe.

Calculated Twice, Same Result

"If quantum gravity in a flat space allows for a holographic description by a standard quantum theory, then there must by physical quantities, which can be calculated in both theories - and the results must agree", says Grumiller. Especially one key feature of quantum mechanics -quantum entanglement - has to appear in the gravitational theory.

When are entangled, they cannot be described individually. They form a single quantum object, even if they are located far apart. There is a measure for the amount of entanglement in a quantum system, called "entropy of entanglement". Together with Arjun Bagchi, Rudranil Basu and Max Riegler, Daniel Grumiller managed to show that this entropy of entanglement takes the same value in flat quantum gravity and in a low dimension quantum field theory.

"This calculation affirms our assumption that the holographic principle can also be realized in flat spaces. It is evidence for the validity of this correspondence in our universe", says Max Riegler (TU Wien). "The fact that we can even talk about quantum information and entropy of entanglement in a theory of gravity is astounding in itself, and would hardly have been imaginable only a few years back. That we are now able to use this as a tool to test the validity of the holographic principle, and that this test works out, is quite remarkable", says Daniel Grumiller.

This however, does not yet prove that we are indeed living in a hologram - but apparently there is growing evidence for the validity of the correspondence principle in our own .

Explore further: Who cares about the fourth dimension?

More information: Entanglement Entropy in Galilean Conformal Field Theories and Flat Holography, Phys. Rev. Lett. 114, 111602 – Published 19 March 2015, dx.doi.org/10.1103/PhysRevLett.114.111602

Related Stories

Who cares about the fourth dimension?

February 3, 2009

Austrian scientists are trying to understand the mysteries of the holographic principle: How many dimensions are there in our universe?

New work gives credence to theory of universe as a hologram

December 13, 2013

(Phys.org) —In publishing a story regarding work reported by Japanese physicists last month, Nature News has set off a bit of a tabloid firestorm by describing an obscure bit of physics theory as "the clearest evidence ...

Mapping the road to quantum gravity

April 23, 2014

The road uniting quantum field theory and general relativity – the two great theories of modern physics – has been impassable for 80 years. Could a tool from condensed matter physics finally help map the way?

Recommended for you

57 comments

Adjust slider to filter visible comments by rank

Display comments: newest first

jalmy
1 / 5 (8) Apr 27, 2015
The 2 dimensional thing happens because gravity and light work in straight lines. Which would be one dimensional but they are also waves. So 2 dimensions.
Tangent2
1 / 5 (5) Apr 27, 2015
To test this hypothesis, gravitational theories have to be constructed, which do not require exotic anti-de-sitter spaces, but live in a flat space. For three years, he and his team at TU Wien (Vienna) have been working on that..


So they created a theory to make their hypothesis work in flat space-time and claimed this to be proof that the holographic principle works in flat spame-time. Now they have to prove that their gravity theory is correct for any of their efforts to pay off.
Torbjorn_Larsson_OM
4.4 / 5 (8) Apr 27, 2015
It is ironic that they had to back down to non-relativistic (galilean) field theories to make a first inroad on AdS-CFT in flat space, since the target is to get a new handle on general relativity. Well, flat space happens to be much harder, and it is hopefully a start...

@jalmy: No, it is a toy model because the 3 and 4 D cases are too hard as of yet. (Though they hoped that at least one side of their proof could be extended to higher dimensions.)

The way I think of AdS-CFT is that it generalizes the usual situation for differential equations to spacetime physics. Usually you can have the local diff eq and know that it applies in a region, and so deduce the global solution if you know the boundary conditions. But it applies for one field (say, thermal "field"), not the whole enchilada.

And here you have it constrain a duality. Like when you can look at either currents or voltages in a circuit, because they are dual to each other. But a deep duality.
arom
1.4 / 5 (5) Apr 27, 2015
Up until now, this principle has only been studied in exotic spaces with negative curvature. This is interesting from a theoretical point of view, but such spaces are quite different from the space in our own universe. Results obtained by scientists at TU Wien (Vienna) now suggest that the holographic principle even holds in a flat spacetime.


By the way, it is interesting to note that nowadays we still could not visualize what the curvature of space is; Maybe understanding its physical could help solving the problem ….
http://www.vacuum...=7〈=en
OdinsAcolyte
3.8 / 5 (5) Apr 27, 2015
Gravitational quantum entanglement.
That just opens up all kinds of wild ideas in my strange little mind.
Wow.
What a Universe. A leaf...on a branch and we know not where the branch is...or the tree or the other leaves. I know. Silly. But it is certainly fun. I do not lay claim to the leaf idea. It is an old one.
deleterjoe
1 / 5 (5) Apr 27, 2015
Our planet is a topological insulator, and out Sun the processor.
docile
Apr 27, 2015
This comment has been removed by a moderator.
SuperThunderRocketJockey
3.3 / 5 (3) Apr 27, 2015
I knew it, Arnold Rimmer is God!

Have mercy on this smeghead, oh great Ace of Aces! You pass all astronavs in my heart, Big Man!
jabailo
2.5 / 5 (2) Apr 27, 2015
I have noted (in the Wiki article on the SImulation Argument) that one thing I find fascinating is TV and Movie screens. With only 2D (and only sound and vision) we are able to evoke a full range of emotion and empathy...even hyperemotion. How can this be possible? Shouldn't we as true 3D beings look at at flat screens and guffaw...and say, this is ridiculous. I can't look at this for even a minute..it's not fooling anyone. And yet, we sit glued.
Returners
1 / 5 (3) Apr 27, 2015
How long is a hyper roll of toilet paper if it has a tube diameter of 1 inch and total diamter of 2 inch, and the paper is infinitesmally thin?

Finite area? Infinite length?
Whydening Gyre
4.6 / 5 (9) Apr 27, 2015
There are definitely some pot smoking questions going on here....
Professor Plum
1 / 5 (2) Apr 27, 2015
Is the Universe a Hologram? Is Superman Real? Wait, is that why there are no stars on the lunar landing photos... oh sorry, I forgot we didn't actually go to the moon. I think W.Gyre is onto something.... definitely some pot smoking questions going on here.
PhysicsMatter
1.8 / 5 (5) Apr 28, 2015
The idea of Universe as a hologram is very interesting in itself and constitute brave departure from instilled in us so called fundamental concepts of space-time but what they did has nothing to do with testing it. GR and QM have their serious problems, these are far from fully established theories, especially on borders of their applicability and to compare new strange theory with no experimental background is simply misguided mathematical exercise. It could be million theories describing something we do not rally understand.

Interesting take on theory v.s. reality I found at: https://questforn...-quanta/
Soinium
1.5 / 5 (2) Apr 28, 2015
If the Universe is a 2D membrane and the multiverse is a stack of parallel universes then quantum tunneling of gravity between 2D universes could explain the dark matter. It is the gravitational shadow of parallel universes.
antialias_physorg
5 / 5 (7) Apr 28, 2015
then quantum tunneling of gravity between 2D universes could explain the dark matter.

"Quantum tunneling of gravity"?
Throwing a couple of buzzwords together ain't science.
zorro6204
4.7 / 5 (3) Apr 28, 2015
This is something that should have been obvious a long time ago, simply by noting that the amount of information inside a three-dimensional object can be contained on the surface, and varies geometrically, not by a cubic function. If you can describe the universe using two dimensions, then the only support for it being in three dimensions is in the human mind, the way we visually map the world.
snd
4.2 / 5 (5) Apr 28, 2015
So they created a theory to make their hypothesis work in flat space-time and claimed this to be proof that the holographic principle works in flat spame-time. Now they have to prove that their gravity theory is correct for any of their efforts to pay off.

--Tangent2

They didn't claim this to be proof of anything for "their efforts to pay out" (whatever that means?). Scientists working on theoretical physics usually do it out of enjoyment, not because they expect a "payout". And simply presenting research does not correspond to "claiming proof".
Returners
1 / 5 (4) Apr 28, 2015
I don't think it's true that 3d could be represented in 2d, because a 2d space needs some external "code" to convert from a 2d space back to a 3d space, and the 2d space cannot store the "code" it needs.

Our computers are 3-dimensional space, even though circuitry is "2d" (actually fractal above 2d), so you can store a 2d space and the code in the 3d , then convert the 2d space to 3d, but this is only true because the computer is a 3d space to begin with.

A 3d space can contain a 2d space and it's laws, but a 2d space can't contain a 3d space and it's laws. That's nonsense. Especially if it's a vector space, because the 3rd dimension is continuous and infinite, so it can't just be "sliced up" like frames in the 2nd dimension and converted with some code.

I call BS on this version of the Holographic Universe concept.
Uncle Ira
3.5 / 5 (11) Apr 28, 2015
There are definitely some pot smoking questions going on here....


@ Whydening-Skippy. How you are too Cher? I'm still good. Don't mind Returnering-Skippy's weird questions non. He'll be back around with the weird answer to go with them soon enough.
Uncle Ira
3.7 / 5 (13) Apr 28, 2015
P.S. for you Whydening-Skippy. Hooyeei I should have been the psychic for the stars me. See there, while I was answering your question ol Returnering-Skippy was writing the weird answer I predicted.
Whydening Gyre
5 / 5 (6) Apr 28, 2015
P.S. for you Whydening-Skippy. Hooyeei I should have been the psychic for the stars me. See there, while I was answering your question ol Returnering-Skippy was writing the weird answer I predicted.

Hey, Ira. Good to see ya.
Actually, Returners has provided decent input in other threads. Just that ALL the prior comments seemed rather "convoluted" , thereby prompting my comment.
That seems to be the nature of these kind of articles, tho. The authors seem to think they are instilling some kind of deep thought process. they're working too hard to tickle the "imagination
bone".
I mean - what is it a "holograph" OF? Itself?
Of course, I could change my opinion after I meditated on the subject...
TechnoCreed
4.7 / 5 (3) Apr 28, 2015
@WG

There are definitely some pot smoking questions going on here....


This comment of yours is certainly the most sensible comment here so far. Although AA's comment was not bad, he should have generalized... sadly it was only aimed at the preceding comment.

Throwing a couple of buzzwords together ain't science.


Personally, I think it applies just as much to Sponge_Bob_Smarty_OM's comment. This guy, who seemingly writes cogent things, never comes back to defend his point when he is challenged; notice that in the future. That is why I am allowing myself to be condescending to him (I think he is scared to look bad).
TechnoCreed
4.5 / 5 (2) Apr 28, 2015
@Ira
We are here to share ideas on science; those who use tricks to impress should be denounced. Torbjorg would probably be good to write sci-fi scripts in comic books. But, although he uses alot of 'advanced knowledge' he pulls answers out of a hat just like an illusionist and that pisses me off.
TechnoCreed
5 / 5 (4) Apr 28, 2015
To have some basic understanding of the Holographic Principle, here is one of the greatest mind alive; Leonard Susskind. He have another quality, he can make complex ideas understandable; he is a passionate teacher.
https://www.youtu...l3Hfh9tY
Protoplasmix
5 / 5 (4) Apr 28, 2015
I don't think it's true that 3d could be represented in 2d, because a 2d space needs some external "code" to convert from a 2d space back to a 3d space, and the 2d space cannot store the "code" it needs.
We make a hologram using a coherent beam of light for the 'code', first to make the hologram of an object, and again to reconstruct the 3d image of the object from the hologram, which is a blurry 2d interference pattern.
A 3d space can contain a 2d space and it's laws, but a 2d space can't contain a 3d space and it's laws. That's nonsense.
Seems that way when you don't understand the math/physics for it, and fail to appreciate what the "C" stands for in CFT.
docile
Apr 28, 2015
This comment has been removed by a moderator.
TechnoCreed
5 / 5 (1) Apr 28, 2015
Still without single experimental evidence, practical usage the less. This is how the physics serves as a salary generator today. Just compare it with cold fusion, which is of immense practical importance, it has thousands of experimental confirmations - yet didn't penetrate the mainstream journals. Are you still asking, how to reform the contemporary science?

Is that so docile puppy dog?
Whydening Gyre
5 / 5 (1) Apr 28, 2015
... and fail to appreciate what the "C" stands for in CFT.

All these acronyms... What does the "C" stand for? As long as your at it, how bout "F" and"T", too?
richardwenzel987
1 / 5 (1) Apr 28, 2015
I always naively believed that any space could be mapped into any other space of higher or lower dimensions. The only exceptions would be spaces that contained the equivalent of a torus or were themselves a torus. Come to find out, there's a lot more to it.
Protoplasmix
5 / 5 (1) Apr 28, 2015
All these acronyms... What does the "C" stand for? As long as your at it, how bout "F" and"T", too?

Conformal Field Theory. Also helpful to know about Conformal mapping, and considering the significance of the work showing the correspondence principle is valid in flat spacetime, it couldn't hurt to wrap your mind around the representation of an infinite amount of asymptotically flat spacetime in a 2d diagram of finite size, i.e., a Penrose diagram :)
Whydening Gyre
5 / 5 (2) Apr 29, 2015
Thanks, Proto.
Will do a little "delving" into it...
TechnoCreed
5 / 5 (1) Apr 30, 2015
@Sponge_Bob_Smarty_OM
4.2/5 for such an idiotic comment...
It is ironic that they had to back down to non-relativistic (galilean) field theories to make a first inroad on AdS-CFT in flat space, since the target is to get a new handle on general relativity. Well, flat space happens to be much harder, and it is hopefully a start...
AdS-CFT Is the concordance between anti-deSitter ( A mathematical model of space of n dimensions with a NEGATIVE cosmological constant ) and CFT ( A 2D Lorentzian scale invariant space ) it comes out from string theory and it synthesize our mathematical knowledge of black holes (from event horizon and beyond) ONLY. Do we need to 'make the inroad on Ads-CFT' as you said? NO, IT IS A GIVEN. It came about when physics theorists were looking for an answer to the 'information paradox'. Of course the Holy Grail of string theorist would be to link GR with QFT but this paper is NOT about GR at all.
Cont.
TechnoCreed
5 / 5 (1) Apr 30, 2015
Now this one is presenting a model of AdS-GCFT but with AdS radius at infinity (odd to think that an infinitely big black hole could be used to describe flat space; the idea is that an infinitely big black hole would not have any space curvature, like our very own big universe). Its only goal is to develop a quantum field model of flat space holography. This model being Galilean makes it valid only for very local immediate space.
Cont.
TechnoCreed
5 / 5 (1) Apr 30, 2015
Even when you reply to a stupid comment you are wrong.
@jalmy: No, it is a toy model because the 3 and 4 D cases are too hard as of yet. (Though they hoped that at least one side of their proof could be extended to higher dimensions.)
NO. It is ONLY about a 2D flat space holographic model. HOLOGROPHY = 2D.
Cont.
TechnoCreed
5 / 5 (1) Apr 30, 2015
The way I think of AdS-CFT is that it generalizes the usual situation for differential equations to spacetime physics. Usually you can have the local diff eq and know that it applies in a region, and so deduce the global solution if you know the boundary conditions. But it applies for one field (say, thermal "field"), not the whole enchilada.
Blaber, blaber, blaber, blaber; the same kind that Zephyr does but somehow, you manage to impress the gallery.
Cont.
TechnoCreed
5 / 5 (1) Apr 30, 2015
And here you have it constrain a duality. Like when you can look at either currents or voltages in a circuit, because they are dual to each other. But a deep duality.
Is this an attempt to present an analogy? How inappropriate.
Cont.
TechnoCreed
5 / 5 (1) Apr 30, 2015
You want to challenge that? Be my guest… here is a link to the ARXIV preprint: http://arxiv.org/...89v1.pdf and a presentation of it: http://www.imsc.r...gchi.pdf
bbbbwindows
2.3 / 5 (3) Apr 30, 2015
Holographic model resolves many current mysteries in cosmology:
Time began..."boot up"
Dark matter...no particles ever detected, null processing
Planck length/time...pixels
Malleable time..only in a virtual reality
Curved space...only in a 3d processing network
Max speed....C is the fastest transmission rate (pixels/second)
Double slit...set of reboots explains quantum waves and quantum collapse
Electron tunneling..quantum world is real, physical world is it's product
Quantum entanglement...code reallocation ignores distance
Anti-matter...reverse processing, anti-time only in a virtual reality
Randomness...client-server acts to which we are not privy

The fact that "physical reality' is mostly empty space and all matter is basically pure energy is the reason this hypothesis is most attractive.
As time goes on we will see that everything, even gravity is an electromagnetic entity. The underlying 2D electric field of limitless possibilities is the "real" universe.
bbbbwindows
3 / 5 (2) Apr 30, 2015
P.S. It also explains how the most powerful information storage system we know of, DNA, supposedly created itself. This has never made sense
EWH
5 / 5 (2) May 02, 2015
Conformal Geometric Algebra (CGA), a computer graphics technique based on Clifford Algebra, adds two dimensions, (one positive-square, one negative-square) to a euclidean space (typically two or three positive-square dimensions) to allow representing circles, spheres, lines and planes as points in the higher-dimensional space. This is done by adding and subtracting the two orthogonal extra dimensions together to make an "origin" (similar to homogenous coordinates, allowing subspaces offset from the origin) and a "point at infinity", which all lines and planes pass through (which distinguishes them from circles and spheres.)

Finding intersections, unions and other operations between the different objects is much easier using this scheme. The 2D euclidean CGA is equivalent to the Minkowski space of relativity, that is, with a (+++-) signature. The 3D CGA is equivalent to adding an additional dimension of the same sign as the spatial dimensions to represent "proper time" (++++-).
EWH
5 / 5 (2) May 02, 2015
2 + 2D CGA is perfect for representing the circular wave crests and troughs of a 2D hologram. In translating from Minkowski space to 2D CGA, the 2D space is the two dimensions perpendicular to the direction of travel, while the sum of the direction of motion plus time is the "origin" and the direction of motion minus time is "infinity". These correspond to the future and past light cones respectively, and are thus "null-square" dimensions. By adding increasing amounts of the "infinity" dimension, circles of larger and larger radius are produced, corresponding with older circular wavefronts spreading out farther from a point source.

Despite this explaination being nearly incomprehensible, CGA is not hard to use practically. The free program "GAViewer" and its tutorials are actually quite intuitive - any one interested in getting a shortcut to understanding a type of math used to describe all areas of physics including quantum mechanics and relativity should try it out.
robandlezlye
4 / 5 (4) May 02, 2015
"...I don't think it's true that 3d could be represented in 2d, because a 2d space needs some external "code" to convert from a 2d space back to a 3d space, and the 2d space cannot store the "code" it needs..."
--Returners

...what about a Fourier Transform? -- the 'code' is in your brain.
gkam
3 / 5 (4) May 02, 2015
I am sure this is true:

There are some politicians and televangelists I can see completely through.
Protoplasmix
5 / 5 (2) May 02, 2015
CGA ... - any one interested in getting a shortcut to understanding a type of math used to describe all areas of physics including quantum mechanics and relativity should try it out.
whoa, awesome, tyVm, EWH, much obliged :)
MRBlizzard
not rated yet May 02, 2015
Here you go. Grumiller's arxiv
http://arxiv.org/abs/1410.4089
viko_mx
1 / 5 (1) May 03, 2015
If the universe is located in the belly of a teddy bear?
Whydening Gyre
5 / 5 (2) May 03, 2015
If the universe is located in the belly of a teddy bear?

No. The Universe IS the teddy bear.
WE are located in the belly...
jsdarkdestruction
5 / 5 (4) May 03, 2015
If the universe is located in the belly of a teddy bear?

Interestingly enough that idea and the existence of a god have the same amount of evidence for them. None. Yet both are technically possible.
viko_mx
2.3 / 5 (3) May 03, 2015
Then I will try to send a message to big Teddy to change his diet and reduce the consumption of dark energy and matter because lately became very darky in modern science.
However I have more confidence in the theory of Big Teddy than the theory of big bang.
FainAvis
5 / 5 (1) May 03, 2015
Is the universe a hologram?! The question is more fitting in a Dr Who episode, where any trite answer can serve as denouement.
TechnoCreed
5 / 5 (2) May 03, 2015
@FainAvis
Is the universe a hologram?! The question is more fitting in a Dr Who episode, where any trite answer can serve as denouement.
Why not listen to good old Lenny Susskind to hear what he has to say about that. From him the answers are never trite and the denouement always very original. https://www.youtu...T357ofuE
bbbbwindows
not rated yet May 03, 2015
Docile: I can't remember where I read it, but there was a very good paper where they have detected the holographic "noise". Their findings led the authors to conclude that the holographic theory was indeed possible.
bbbbwindows
5 / 5 (2) May 03, 2015
The most compelling evidence from a common sense view is that all " matter " is mostly empty space and what we perceive as solids are just energy force fields. There is convincing neuroscience research that demonstrates our brains are not only receivers but actually "projectors" of the energy which we perceive (i.e. the hologram). The dozen points I made earlier that demonstrate the usefulness of the theory in explaining current "mysteries" coupled with this type of research indicate that this theory may in fact be correct. But it is still a hard one to accept.
OdinsAcolyte
not rated yet May 04, 2015
I suppose flat could be as relative as time.
It would depend upon the point of reference...
malapropism
not rated yet May 04, 2015
Is the universe a hologram?! The question is more fitting in a Dr Who episode, where any trite answer can serve as denouement.

Hey, don't denigrate Dr Who.
Mayday
not rated yet May 04, 2015
Well then, the Easter Egg hunt begins. Somewhere in the the universe there certainly must be hidden a miniature hologram of the hologram. Or, being a hologram, isn't it hidden in plain sight?
Mike_Massen
3 / 5 (2) May 27, 2015
viko_mx mutters laconically
Then I will try to send a message to big Teddy to change his diet and reduce the consumption of dark energy and matter because lately became very darky in modern science. However I have more confidence in the theory of Big Teddy than the theory of big bang
Your big loving teddy viko_mx that doesn't act like ANY lover, is a devil punisher with ONLY source of idea being a book by men :-(

Whereas big bang theory has support of physics/maths independent of any singular idea as a claim in a book but, based on foundation of Physics & Maths is attainable without ANY expectation of claims/faith in a book which are ALL untestable. ie ALL religions are based on claims which CANNOT ever be tested, the claimed deity is ALWAYS silent !

Why is that viko_mx ? why are ALL deities silent, is it somehow consistent with them ONLY being claim ?

Just like your 'big teddy' manufactured from the ideas of men, some sincere, some insecure & all ego based !

Please sign in to add a comment. Registration is free, and takes less than a minute. Read more

Click here to reset your password.
Sign in to get notified via email when new comments are made.