|
| If this is your first visit, be sure to check out the FAQ by clicking the link above. You may have to register before you can post: click the register link above to proceed. To start viewing messages, select the forum that you want to visit from the selection below. |
|
|||||||
| Register | FAQ | Members List | Calendar | Mark Forums Read |
![]() |
|
|
LinkBack | Thread Tools | Display Modes |
|
|||
|
Quote:
Quote:
And how do your EOS calculations and equations differ from the EOS calculations and equations (see appendix A and B) here?
__________________
Some try to tell me, thoughts they cannot defend,... - Moody Blues. Neptune- The original Dark Matter. The author feels that this technique of deliberately lying will actually make it easier for you to learn the ideas. - Donald Knuth |
|
||||
|
I could not refer to the site you posted for some reason or another. The singularity I am speaking of is the same for that proposed for the Big Bang and black holes. The equations are simple. If a body must be squeezed down to zero dimensions in order to produce a singularity, then its boundary must have a zero radius. We can therefore find the relationship between the body and that of a particle that exists on this boundary to see if it is possible to squeeze the body down indefinitely. The total energy of the body divided by its volume gives us the energy density, which is the internal pressure. Multiplying this by the area of a particle on its surface gives us a force pushing outward, or F=[Ebody/(4pi/3)R3]Apart. The gravity between the body and the particle will also create a force directed inward, or F=GMbodyMpart/R2. But the outward force is inversely proportional to the cube of the radius while the gravitational force is only proportional to the square of the distance, so that it increases much more rapidly for a smaller radius. If the body were to decrease in size ten-fold, the gravitational force would increase by one hundred, but the internal pressure increases by a thousand.
I know that some of you might argue that the area of the particle would also decrease in proportion, so that the outward force would be much smaller. This is debatable and I would like to hear your comments. It is my contention that the areas of particles is what determines the gravitational force as well, so if the areas of particles were to decrease, then the gravitational force would decrease along with it. It is also said that a singularity might create such anomalies as wormholes. Even if it were possible to create a singularity, how one gets a wormhole from this I don't know. If one likes the idea of sliding through space and time in wormholes like waterslides at the funpark, then far be it from me to spoil the excitement. Go ahead, have fun with it. But if one wishes to deal with real concepts of science, with real consequences and conclusions, then I'm sorry. The ride is closed. |
|
|||
|
Quote:
What you are saying 'might' apply to a stellar black hole singularity. Tim Thompson indicated that there was a solution for a stellar balck hole where the singularity was actually the accretion disc, at least I think I remember that correctly. Think about this...we know where the mass comes from for the stellar black hole, right? The 'mass' of the star determines the 'mass' or GRAVITY of the black hole. Where does the 'mass', in other words, GRAVITY come from to make a 2 billion sol mass of a SMBH??? [The total energy of the body divided by its volume gives us the energy density, which is the internal pressure] This assumption is definitely a problem for a SMBH and the singularity within!
__________________
RussT ________________________________ Everything is, as it should be, otherwise, it wouldn't be! |
|
|||
|
Quote:
Putting this another way, the physical regimes in which quantum theory and GR are so strongly mutually incompatible as to render any statements about the nature of stuff in them quite meaningless are well known, and such regimes 'occur' long, long 'before' any singularities are reached. Quote:
Quote:
If so, then as (AFAIK) no one has done any experimenting or observing inside a BH's event horizon (or at least they haven't been able to communicate the results of such to us), or during the first ~10-43 seconds of the universe's existence (etc), all this beautiful theorising is equivalent, scientifically, to speculation, isn't it? OTOH, if someone comes up with a theory that has consequences that are observationally/experimentally testable (even if only in principle), then we're into a whole new ball-game, right? |
|
|||
|
There is something troubling about the idea of applying a theory or a set of theories to any specific time after an earlier specific time that cannot be observed. Can a theory validly claim to say that there is a set of conditions that apply, say, 100 Planck times later than 1 Planck time, without invoking the first Planck time? How do the "100 Planck time" conditions manage to ignore the first Planck time?
|
|
|||||||
|
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
My biggest comment is that you are not using the correct equations. This idea of yours sounds as if it is a combination of some college physics and popular science simplifications of General Relativity. It does not sound like the idea has been fleshed out, using the correct physics or equations. This paper give you an overview of EOS equations and their solutions. What do you propose is incorrect about these equations, that we should use the equations you posted instead This site models rotating neutron stars. Quote:
__________________
Some try to tell me, thoughts they cannot defend,... - Moody Blues. Neptune- The original Dark Matter. The author feels that this technique of deliberately lying will actually make it easier for you to learn the ideas. - Donald Knuth |
|
||||
|
Quote:
Quote:
Quote:
Quote:
__________________
Some try to tell me, thoughts they cannot defend,... - Moody Blues. Neptune- The original Dark Matter. The author feels that this technique of deliberately lying will actually make it easier for you to learn the ideas. - Donald Knuth |
|
|||
|
Quote:
IMHO, the best you can ask of good theories is that they are internally consistent, consistent with other (good) theories where their domains of applicability overlap, and consistent with (all, good, relevant) observational and experimental results. From there, you can say "IF {theory} applies in {domain you are interested in}, THEN {conclusions}." That conditions may have been different 'before' is not at all unusual in science, and (mostly) irrelevant. For example, you don't need to know how the universe began to test the validity of the (economic) theory of comparative advantage, or Darwin's theory of evolution (as it applies to life on Earth). |
|
|||
|
Quote:
I've noticed that sometimes links to PDFs cause Firefox to hang (so I've learned to sacrifice IE for these purposes), and that they can be very, very slow loading if you're on a low bandwidth connection. Perhaps you could right-click on "Save Link As" (or whatever your browser says)? |
|
||||
|
Thank you, Nereid. it worked that time when I saved it. The document seems rather long and complicated, however, so it may be some time before I complete it. In the meantime, let me throw these thoughts at you all. First of all, it would indeed be very difficult to speculate on the workings of an environment in which we cannot even see past the event horizon. But isn't the main concept of a black hole really only that light cannot escape its confines? Granted that would provide room for some strange effects within its boundaries, effects that may not agree with everyday experience, but shouldn't most of our physical laws still apply to much the same degree as outside. For instance, if a star were to suddenly collapse into a black hole, its gravity outside of the event horizon shouldn't change at all. Just because we can't "see" what is going on doesn't mean that it is anything so extraordinary that we should abandon our laws or even common sense.
It is speculated that the laws of physics might change dramatically within a black hole, so that even space and time are influenced. But time is a man-made concept, used as a constant of proportionality for change. It is simply a comparison of the rate of change of one event to another very stable and reliable frame of reference, such as the rotation of the Earth or a clock. We must follow this ideal definition as precisely as possible for it to be of any use to us. Therefore, time cannot slow down or speed up, go backwards, or sideways to alternate universes, or upside-down. Space has only three dimensions as it is defined. This is by the minimum number of lines necessary to identify the position of a specific point from a point of origin, which comes to three when they are directed perpendicularly to each other. It can be said to be two when it comes to a surface, such as the surface of the Earth, requiring only a longitude and a latitude. But this is an illusion. One of the dimensions has already been identified, stretching from the north pole to the south pole as an additional frame of reference, or "line" of origin, which is the Prime Meridian, and from this the longitude stems. Even a sheet of paper has three dimensions, but one of the dimensions is constant and irrelevant, so that we only consider the other two. But if we were to bend this sheet of paper in any way, or to curve it upon itself, then we must further consider the definition of this curve, and give it back its third dimension. So to consider that two far away points can be made to exist in the same place at the same time by the curvature of space, as in the case of wormholes, is ludicrous. It is but a parlor trick. If the universe began as a singularity, then it must necessarily exist within another universe of greater spatial dimensions than what we know. What then, of singularities within our own universe? Would they then lead to lesser dimensions? Last edited by grav; 29-May-2006 at 01:26 AM.. |
|
|||
|
Quote:
attempts to try and use a worm hole for something other than it was intended!!! Just like your treatment of time above, "ALL" the Sci-Fi HAS to be taken out of these discussions, if we are to figure out how GR really does apply to the physical workings of our universe!!!
__________________
RussT ________________________________ Everything is, as it should be, otherwise, it wouldn't be! |
|
|||
|
Quote:
Originally Posted by RussT What you are saying 'might' apply to a stellar black hole singularity. Tim Thompson indicated that there was a solution for a stellar balck hole where the singularity was actually the accretion disc, at least I think I remember that correctly.] Quote:
Quote:
why math? The real problem here, ladies and gentlemen, is that the T=0 singularity is only a "PROPOSED", "POSSIBLE" singularity, where-as you Guru GR specialists have now determined that the SMBH and it's singularity "actually exists". That is a tremendous human achievement!!! Now there are 2 keys to understanding the rest of this. The first is to understand that the singularity in the SMBH, DOES NOT suffer the same 10 ^-43 failure as the cosmic one, because it is not trying to start the universe!!! In fact, it can be, and has been modeled all the way to, and through the singularity!!! From the outside going into the SMBH... accretion disc/event horizon/black hole/ring singularity/worm hole/white hole That is the absolute beauty and elegance that the maths of Einstein's General Theory of Relativity, when applied correctly, physically descibe part of the processes of our universe! And yes, Nereid, I do mean physically! Now, a really astute, intuitive GR learned person, might have figured out, that ALL GR singualrities MUST be affiliated with a black hole, and since it has been definitely determined that our universe is not "IN" a black hole, therefore the PROPOSED singularity could not exist, therefore there is no 10 ^-43. Then they may go one step further, and realize that since the proposed singularity had modeled the nucleosynthesis so well, that maybe, just maybe, the "REAL" singularity, you know, the one in the SMBH, could be modeled to do exactly the same thing! The key to this is understanding a couple of defintions that have been slightly misunderstood. One of those is Tim's use of Infinite Curvature...in the SMBH it is the tendancy toward infinite curvature, but it doesn't go all the way to infinity, otherwise they couldn't have modeled the "Ring Singualrity", right Tensor? The other defintion that needs to be understood, is "White Hole". There are two, which has been one of the reasons, that this whole concept has remained so SCI-FI, but once you UNDERSTAND each one seperately, it truly does apply physically to our universe!!! To See How This Can Be, since this is straight GR, Go To Astronomy>"The Birth Of A GAlaxy"
__________________
RussT ________________________________ Everything is, as it should be, otherwise, it wouldn't be! |
|
|||||||||||
|
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
__________________
Some try to tell me, thoughts they cannot defend,... - Moody Blues. Neptune- The original Dark Matter. The author feels that this technique of deliberately lying will actually make it easier for you to learn the ideas. - Donald Knuth |
|
|||
|
Quote:
Look here and explain to me just what equations in there are ludicrous or a parlor trick. Quote:
__________________
Some try to tell me, thoughts they cannot defend,... - Moody Blues. Neptune- The original Dark Matter. The author feels that this technique of deliberately lying will actually make it easier for you to learn the ideas. - Donald Knuth |
|
|||
|
Grav is right about a singularity.
A singularity was a simple hypothesis a time ago. An object above a star has its potential gravitational energy. This potential energy may be transformed in the kinetic energy which alows to go through a star and reach the same potential energy again as previous. It is the energy conservation law. A Black Hole is a closed energy system. The Black Hole do not allow to lose the energy as it is possible in a radiating star. In the Black Hole all objects exchange the energy between each other. If there is a friction it excites the next objects. The energy can not escape from a Black Hole. In real Black Hole like object is there more energy input then output. Every infalling object has its kinetic energy + potential energy of the BH gravity. The energy can not be separated from a rest mass and the particles have their kinetic energy always. They oscillate, rotate, move. This real Black Holes is called Gravastar. There is not an overdense singularity possiblke. There is gravity balanced by the kinetic energy pressure. |
|
||||
|
A singularity only occurs in the equations, IMHO, when we do not know the physics that is coming into play, in gravitation. I strongly doubt that all the mass of a BH is concentrated in a 0D point.
I am sure this "problem" will be resolved at some point. When we get a good theory that describes the physics of matter beyond what is possible at this time, we will be able to describe what the concentration of the mass and what its dimension and size is in a BH. Like I have said before, a singularity in your equation does not mean that there is nothing there. Take a look at plasma wave dispersion relations, full of singularities, and luckily so, otherwise we would not have any waves.
__________________
Any comments in glorious red are to be considered in ModeratorMode. 善數, 不用籌策 (shàn shù, bù yòng chóu cè) He who is good at counting, uses no counting tools “A good scientist has freed himself of concepts and keeps his mind open to what is” 道德經, 二十七 (dào dé jīng, 27) |
|
|||
|
Quote:
__________________
Some try to tell me, thoughts they cannot defend,... - Moody Blues. Neptune- The original Dark Matter. The author feels that this technique of deliberately lying will actually make it easier for you to learn the ideas. - Donald Knuth |
|
|||
|
From Wikipedia: “Physical cosmology, as it is now understood, began with the twentieth century development of Albert Einstein’s theory of general relativity and better astronomical observations of extremely distant objects. The twentieth century advances made it possible to speculate about the origins of the universe and allowed scientists to establish the Big Bang as the leading cosmological theory, which most cosmologists now accept as the basis for their theory and observations.”
If general relativity theory applies in the evolution of the universe, it must not be a good theory, since it is not in agreement with other good theories where their domains of application overlap (since general relativity predicts an initial gravitational singularity), and it is not consistent with all good observational results (although the fact that it predicts its own demise in an original gravitational singularity may save it). Yet is it not true that the assumption that the universe began in a hot dense state (“the Big Bang theory”) is based on the emergence of the universe from the predicted gravitational singularity of general relativity? Is it good to base a field of science on the assumption that the universe began in a hot dense state, to teach generations of students based on this assumption, and to study the currently observable universe based on this assumption? It seems to me that knowing how the universe began is a fundamental goal of physical cosmology, since cosmologists appear keen to speculate upon it. Nereid wrote, “That conditions may have been different 'before' is not at all unusual in science, and (mostly) irrelevant. For example, you don't need to know how the universe began to test the validity of the (economic) theory of comparative advantage, or Darwin's theory of evolution (as it applies to life on Earth).” Would a cosmologist say that it is, or is not, necessary to know how the universe began in order to test the validity of Big Bang theories? |
|
||||||
|
Quote:
However, there are other domains of applicability which overlap. For example, low mass-energy density. In these domains, GR (in its 'reduced' form, SR) and quantum theory (in its reduced from, considering only the electro-magnetic force) combine to produce the most stunningly successful theory of science, of all time ... QED. Quote:
Quote:
If all that you seek is a model, based on some theory or theories, which can account for the observational results that you have to hand, then it is irrelevant what happens (significantly) 'before' the things being modelled. If you are looking for some philosophical generalisation of the (limited) science, then the question you ask becomes (very) important. [nitpick]that 'the universe began in a hot dense state' is not really an 'assumption'. It's more 'IF you plug in {hot dense state} 'initial conditions' into {models which incorporate GR and Standard Model particle physics and the behaviour of plasmas and ...}, THEN you get {model outputs} which closely match what you observe, through telescopes.[/nitpick] Quote:
Why not start a new thread, in General Science, on this topic? Quote:
Quote:
General comment: the Nereid shorthand for the 'three consistencies' criteria for theories is a sound bite summary. As ngeo has shown, in the preceeding post, that summary needs fleshing out, at least in some cases (good topic for a new thread in General Science?) |
|
||||
|
Quote:
|
|
|||
|
The OP seems to contain a number of misunderstandings, or confusions, concerning 'singularities'.
These have been addressed, in subsequent posts. This thread subsequently seems to have moved away from the OP, into the contents of science teaching (for example). I am having difficulty working out just what ATM idea has been proposed (other than a misunderstanding of some aspects of modern physics and cosmology) - can someone please state the ATM idea(s) that are being defended? If there are no ATM ideas under discussion, is there any reason why this thread should be kept open? |
|
|||||
|
Yes, there are reasons to keep this thread open.
I fully expect Tensor to defend all of his statements above, and I fully intend to defend mine, or recant on certain issues, if need be, and would hope that he is capable of the same. Quote: Originally Posted by RussT The real problem here, ladies and gentlemen, is that the T=0 singularity is only a "PROPOSED", "POSSIBLE" singularity, where-as you Guru GR specialists have now determined that the SMBH and it's singularity "actually exists". That is a tremendous human achievement!!! Quote:
So I am not saying that GR is actually saying what is happening here, just that GR has identified they exist! [There is nothing more to a singularity], Tensor, this is just an opinion! My opionion is, that they have a physical reality in our universe, just as much as E=MC^2 was shown to have a physical realty in making the Atom bomb(which Einstein seriously doubted for months, until his assistant convinced him otherwise) Quote: Originally Posted by RussT The first is to understand that the singularity in the SMBH, DOES NOT suffer the same 10 ^-43 failure as the cosmic one, because it is not trying to start the universe!!! Quote:
But let me ask you this...If we could take a SCI-FI trip together (imagine that conversation), and we could be in a ship that could travel out side our universe and back in time 13.7 billion years in the past, and we could actually see the proposed Big Bang, Friedmann Naked Singularity, as it happened and then Inflated the universe, what would you then say about the Planck era 10 ^-43 seconds??? Originally Posted by RussT Now, a really astute, intuitive GR learned person, might have figured out, that ALL GR singualrities MUST be affiliated with a black hole, Quote:
Notice in my quote above, I very purposely said "Affiliated with" and not inside, as you did! And since you are directing me to a site that you seem to be implying says that singularities can be 'not inside' black holes, does that mean that they have finally identified that elusive and much talked about and bet on "Naked Singularity"? And yes, I understand Cosmic Censorship. Speaking of the bet, here are two telling lines from a New York Times "QUOTE"... Quote:
But to be fair and correct, Dr Preskill went on to say... ''If we are ever to understand singularities, we must do so in terms of some yet-to-be-discovered theory of quantum gravity, and that would be a revolution in physics. We're not there yet.'' And I agree. Quote: Originally Posted by RussT Then they may go one step further, and realize that since the proposed singularity had modeled the nucleosynthesis so well, that maybe, just maybe, the "REAL" singularity, you know, the one in the SMBH, could be modeled to do exactly the same thing! Quote:
So to clarify...[First off, the BB singularity is of a different type from black hole singularities. At the BB singularity, all world lines in GR start there. In a black hole (size doesn't matter), all world lines end.] You know that I know this, we've been through it, and I have used it in examples to others! Yes, the BB singularity, is a Naked singularity, and all the world lines start there, and the singularity in the Super Massive Black Hole is where they end.WOW, do you see what just happened??? In this very last sentence? The Bold?
__________________
RussT ________________________________ Everything is, as it should be, otherwise, it wouldn't be! |
|
||||
|
Quote:
But as far as bodies that travel close to the speed of light, even the special theory of relativity really only implies that only that which travels at this speed will slow down their "atomic rate of change", if you will, not the entire rest of the universe, which is what is required for time to slow down or go backwards. Since time is constant by definition, it is only the "atomic rate of change" for that which travels close to the speed of light which can slow down, not time itself. This would make the speed of light, not a time portal, as seems to be the formal consensus, but more of a fountain of youth. As far as singularities go, I think people are intuitively realizing the complexies and problems for an R=0 singularity since theories of an oscillating universe are also in circulation, where the universe will collapse only to the point where internal pressures overcome gravity at R>0 and then expand again. But if this is the case, wouldn't it be just as simple to imagine this within three dimensional space? However, the idea of an oscillating universe has also recently been dashed by an accelerating expansion. It is my contention that most of the redshift is caused by space friction of light by neutrino pressure, where fobserved=femitted/[eHo*distance/c)]. One will notice that for a small distance, this approximates Hod/c as predicted. But for larger distances, the reshift seems smaller than it should otherwise be, which is the reason for the accelerating universe hypothesis. Quote:
Last edited by grav; 30-May-2006 at 03:09 PM.. |
|
|||||||||||||||
|
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
Quote:
__________________
Some try to tell me, thoughts they cannot defend,... - Moody Blues. Neptune- The original Dark Matter. The author feels that this technique of deliberately lying will actually make it easier for you to learn the ideas. - Donald Knuth |
|
||||
|
Quote:
Relativity says that the traveller will not age as much as the non-traveller. That seems to be the case as time dilation has been observed. (Ref. GPS satellites.) Quote:
Quote:
How do you define time? Our current definition of a second is the duration of 9,192,631,770 oscillations of cesium atoms excited by microwaves. That is a rate of change at the atomic level, which would seem to mean that time is atomic rate of change. (Edit: Star, not Start. Ergh.)
__________________
Never attribute to malice what can be adequately explained by ignorance or stupidity. Isaac Asimov |
|
|||
|
Re GR being inconsistent with observation, the first example that comes to mind where general relativity is not consistent with observations is in a recent report of a kind of gravity being produced in a laboratory experiment. As I recall, the effect is many orders of magnitude greater than that predicted by GR.
The second example is dark energy. As I understand it, in GR dark energy is the cosmological constant Lambda, also termed vacuum energy, and is 120 orders of magnitude smaller than the quantum vacuum energy observed as the Casimir effect, with “no known natural way to derive the infinitesimal cosmological constant observed in cosmology from particle physics.” (Wikipedia) If it is argued that Lambda “explains” the observation which it is intended to, namely apparent accelerated expansion (which observation is also subject to differing interpretations and is suspect to many people), and that cosmology must move beyond particle physics, it seems to me that is ignoring what is already observed in favor of what is not only not observed but also unknown. (However this is all really semantic argument by me, and the difficulties are far better understood by people who know GR.) Without dark energy (and dark matter), quoting from an assistant professor at MIT: “General relativity is inconsistent with cosmological observations unless we invoke components of dark matter and dark energy that dominate the universe.” (Sean Carroll) These examples are not why I question GR, which it seems is indeed a very good theory. It is its application to the universe as a whole in Big Bang theories, with the assumption (which regardless of how it is spun stems directly from the GR singularity) that the universe began its existence in a hot dense state. In addition, space-time and matter in GR have a curious physical-nonphysical relationship which obscures the possibility that matter may evolve from energetic space. As far as a General Science thread on the role of science, its relationship to reality, its limits, the role of theory, the importance of good alternatives and so on, and the earlier suggestion about cosmology in particular, I think cosmology suffers simply from its scope and its foundation on two (what seem to me to be) awkward competing theories. Is it legitimate to separate a field called “physical cosmology” from a field called “astrophysics“? What led to it? Maybe that is where a thread could start? I am in no position to figure that out present but it would be interesting to read what people say. |
|
||||||
|
Quote:
Quote: Originally Posted by RussT But let me ask you this...If we could take a SCI-FI trip together (imagine that conversation), and we could be in a ship that could travel out side our universe and back in time 13.7 billion years in the past, and we could actually see the proposed Big Bang, Friedmann Naked Singularity, as it happened and then Inflated the universe, what would you then say about the Planck era 10 ^-43 seconds??? Quote:
"IF', "IF", "IF", we could see it as it happened and witnessed the cooling, etc, what would we say about the importance of the Planck time/length/size??? Quote:
Quote:
Quote:
Quote:
Originally Posted by RussT Yes, the BB singularity, is a Naked singularity, and all the world lines start there, and the singularity in the Super Massive Black Hole is where they end.WOW, do you see what just happened??? In this very last sentence? The Bold? The reason I stopped my clarification right here, and boldened this sentence, is because, it is the answer! Just take the BB singularity out (you said above that it was not proposed), and do a lookback (Time reversal, perfectly valid solution in GR) of the Singularity in the SMBH. Just start hauling it up from the depths of the SMBH and eliminating the black hole as you go. This, btw, is the same exact lookback scenario when you shrink the universe to a point, or near point, since you want me to be so precisely correct. Now, when you get that SMBH singularity above the event horizon, WHAT DO YOU HAVE???
__________________
RussT ________________________________ Everything is, as it should be, otherwise, it wouldn't be! |
|
||||
|
Quote:
Quote:
As far as time dilation is concerned, it has been demonstrated with an atomic clock in a jet and with an unstable particle which has been found to take longer to deteriorate when accelerated to a velocity close to the speed of light, and I'm sure other ways as well. I am not arguing against this. It is merely the way we perceive it that I have a problem with. Let me ask you this. If these two experiments which I have just stated are shown to agree with relativity and all of its equations exactly, so that there is absolutely no doubt as to their results (which again, I am not arguing against), which way do you think it would be an easier and more sensible way of thinking about it? 1)That the greater velocity has somehow effected the flow of time so that time itself is variable, or 2)That the greater velocity causes a greater pressure to act on the body which slows its "atomic rate of change" There is a difference here. Quote:
|
![]() |
| Thread Tools | |
| Display Modes | |
|
|