So, the V coordinate is the V minus r coordinate. So, with the center blackboard R by V T will be greater than 0. So, if we have a V amount 0 it is greater than 0. So, you can have all the way that. And the V coordinate lines are possibly greater than 6 lines are constant or this is the line constant. Now, please. What do you say? This is going to be r by V T like how V r by V T along the unicycle like no. So, again max V V by V T is like the velocity that we will be talking about. And sorry V r by V T is usually the velocity that we will be talking about for a part. You are talking about whether something exceeds the speed of light or not? Yeah. I mean V r by V T. You know the question of whether it exceeds the speed of light or not is just a question of whether it goes and stays at its future light. Everything always stays inside its future light. Is different relations between different you know coordinate. But the light code is the geometric concept and everything always stays with it. But the light code can turn like for example, there is a diagram in the non-stick. That's right. The light code can turn in those coordinates. In this picture the light code is always true. Right because that's how light this is an in going light this is not going. The light code is the thing that is that is enclosed between in going and the outside. Now this is because this is a nice well drawn diagram in real coordinates. Now if you draw on some other coordinates the light codes can get it. If we are drawing R P coordinate. You want to draw the light codes in R P coordinates? So that we have enough of you. I will have to think of that. So let's see. So how do the light codes work? A R P coordinates. So this is the question of what do the R P coordinates look like in this system. So see basically this is the point.  And T is the space magnet. Going to smaller R is going to future time. So of course the light code coordinates we can just solve by setting ds by equal to 0.  So this is the light code. Outside. So first let's look at the motion of the light trace. So how do the light trace move? They move by d R by d T. As you put a plus minus 1 minus 2 by m by R. So this was squared. This was squared. So we get the plus minus. Let's... These numbers are negative inside the... Inside the... Inside the... Inside the calculus. Okay. So now the future directly light ray. Always go... So suppose you want to draw the two light codes that are pointing towards the future. In the article suppose I draw R and T here. The future directly light ray is always towards smaller R. Okay. And then it's done. So it's always once when... This is R is equal to m. It's always pointing like this. Okay. And how much T changes is given by this one. So one of them is smaller than one. Whereas outside here, the future direct light code is always towards larger T. Okay. So the light code took me... So here the light code looks like this. Here you can put someone like this suppose. I suppose at this point it's sort of fancy. And here it didn't start. So the numerical value then... If I do the... It would be more than... Yes. The numerical value has no meaning. We only have that... Yes. You have to... You have to... If the question is... Spirit of light is a question of... Is the ray in space light? Not the light. The numerical value changes when it changes. It's got to do. Okay. This is all the orientation. Okay. Okay. That's all. Okay. So this is our last lecture. And there's a lot of things I would like to... If you guys are gay, we could have these extra classes after December 15th on A, the theory of inflation, and B, some more details about that. Okay. I'm related to the cross. But just in this lecture... I have a book called... In this lecture there are many things about light code physics that we should talk about and that I will sort of touch on. I work on all details of everything just so that I can tell you about many things. Okay. The first thing about light codes that I wanted to tell you about was this notion of the eventualism. Okay. So in this solution that we've talked about, in this particular solution that we've talked about, we see that there is this surface and we are equals to m beyond which things look different. I mean, they don't look globally different. They look globally different. Okay. And now you can ask, how do you invariantly characterize this difference? I mean, what is the clear statement? And the difference is this, you see, there are some regions of space time from which a light ray can reach that's its coordinates. But there are other regions of space time from which it can. Okay. So if some regions of space time are causally connected to space time, do you have an observer here? You can receive signals. Or let's say you have an observer here. Some large boxes down. You can receive signals from the sky if it's here, but not from there. Okay. So now when there are some regions of space time that have a property and other regions that don't, there's a boundary between these two. Regions. And the boundary between the regions of points that can communicate with the scribe class. And those that cannot. As a name, it's called the event release. So in this diagram, the vaporizer is the surface. Now this, of course, is a very simple solution. Okay. Sorry. In this diagram, if you really write scribe class in this one, which scribe class is it? Okay. So if you put scribe class prime, then the event release for that would be this. But let's say we choose the vaporized device in this scribe class. That would give this interest. Okay. Because anything to the left of it cannot communicate. Okay. Now, of course, having these two scribe class and so it's an artifact of dealing with this internal domain. As we discussed. Okay. In the tenduous diagram of lack of form, due to colants, which looks like this. There's only one. Okay. There's this confusion of that. There's something. Okay. So this already brings you to a point that the notion of the event release is a little, then this particular static launch. In particular, suppose we have the situation that we talked about at the end of last class, because let's say there's a shell of light that is imploded, or a shell of dust that implodes. Okay. To start off with, space time could be very, you know, near flat everywhere. But this implosion can cause, can cause the creation of black. As you see, it's a tremendous thing. So in this full space time, including the full dynamical space time, with all the dynamical details of the formation of the black hole. Okay. There's still this notion, there's still a separation, the big one, of space time that can communicate with survivors and those that cannot. And there's a boundary between these two points. There's a boundary between the imaginables. So the event horizon is well defined with a complicated dynamical situation that involves black formation or involves scattering of black. Okay. Now, there are many things about the event horizon that are very interesting. The first thing about it is, that is what it's called now the event horizon. So the event horizon is a manifold, it's a sub-manifold of big black space time in four dimensional space time, it's a three dimensional manifold. And, any co-dimension once of a sub-manifold can be, you know, it's conveniently described by some equation of the four measures equal to zero where it's a sub-manifold. Okay. Now, even a manifold like this, there's a notion of the normal to the manifold. So what is the normal to such a manifold? The normal to a manifold is a one-fold. It was composed in such del mu of s. Take the function s, take del mu of s, that gives you one-fold because there's lower matrices. So the normal to a manifold is characterised by one-fold. You can also characterise by vector, by basing, because naturally it's characterised by one-fold. Now, what is the defining property of this one-fold? The characteristic property is that, look, suppose s is equal to zero on this manifold. Then you start from a point on the manifold. You take del mu of s and multiply it by dx mu. So then dx mu stays on the manifold. That takes you to another point on the manifold where s is equal to zero everywhere. So this is equal to zero. Okay? So the normal one-fold is called the normal, because it is orthogonal to all vectors that lie in the tangent space of this manifold, of the sub-manifold. In Euclidean space, the sub-manifolds are very easy to understand. You can cut a normal vector, and now you're enticed to then multiply it. Now, even in the tangent space, something else counterintuitive happens. If counterintuitive, if you're thinking of Euclidean space. And the counterintuitive thing that happens is this one. You see, since we've got a manifold with, let's try to characterise these manifolds. It's clear, since there are three kinds of one-folds, three kinds of vectors, qualitatively three kinds of vectors, you know, one-fold we have a space like, like, like or not. Okay? So it's clear that locally there are three distinct kinds of manifolds. Suppose you've got a space like, suppose you're looking at a manifold with normal and specific. You can always boost the good coordinates in the local little batch so that your manifold is x i equal to 0 in a local, uh, non-existent space. Okay? So the normal is given by tangent. It's given by in the coordinates t, x 1, x 2, and x 3 as the components 1, 0, 0, 0. So one-fold will be the components 1, 0, 0. Okay? So that manifold is like space locally in some way. So to keep it out of the space, it's such that a manifold with a space like locally always looks like that. All right. On the other hand, a manifold could be time like. In some appropriate coordinate system, the coordinates on the manifold could be t, x 1, x 2. The normal could be x 3. So that's a time like that. A manifold where the normal thread is space like, the manifold includes two directions of space somewhere like that. Okay? The most interesting kind of sub-manifold in the red since space, a sub-manifold where the normal is null. Okay? So, uh, just to get some intuitions with this, by appropriate coordinate changes locally, you can always take any null vector and put it in the form 1, 0. 1, 1, 0, 0. Let's see what vectors lie what local, uh, what local tangent vectors lie with this method. So that's everything that's rounded to this. So, of course, 0, 0, 1, 0 and 0, 0, 0, 1 lie with the manifold. But in addition, one more vector lies with the condition of normality. It's one condition of four-dimensional vector space. It gives you three-dimensional vector space. Can you tell me what the third vector is? No. 1, 1. 1, 1, 0, 0. Because this vector is null. So it's in particular orthogonal to itself. Null manifold. The null manifold is spanned by two space lines and one null direction. And the null direction is the normal direction itself. So the null manifold, the normal vector lies in the manifold. So it's normal in the manifold, but also it's an tangent to it. It's really counter-intuitive. I mean, even if you go normal in this direction, it stays on the manifold. In this direction, it stays on the manifold. Okay. Yes, so this is a tangent vector. It's both normal as well as tangent. Sir, is there only one such manifold? One such manifold. Locally, of course. No, but there are many nulls. No, I was saying flat-space. In flat-space, right? Let's see. Basically, any s such that it's gradient is null. Defines an untenable. Okay. So what does that mean? What we want is m or s and m or s to be 0. Now, let's look at it as an example of such a manifold. Yeah. So for instance, suppose we take a manifold that is where s is equal to s of x minus d. This will do it. But if we take s as equal to s of r minus d, that will also do it. Of course, x minus d, y minus d are related to each other. The r minus d is not related to each other. Okay. Now, at least if you're not interested, it's not one of your possible singularities. Two-dimensional polarity. All I have to do is I have to say it. Excellent. Now, why do I tell you all of this? I told you all of this because we just introduced the event horizon. Okay? Now, the event horizon is such a manifold and we don't understand what time it is. Okay? And of course, you know, you know, there's a particularly interesting surface and a particularly interesting kind of manifold. It's going to be a lot of intervals. So what I want to prove to you is that the event horizon of any black hole is obviously not. We'll show this by contradiction. Suppose we were going to append to a black hole. Yes, that's right. Okay. It's almost obvious. Let me prove it. Suppose at some point, the manifold was a space like. I mean, at a space like norm. Okay? So then, some local coordinate system, we could think of this manifold as including time and two special directions. These two special directions are found in this coordinate system. This time is the one direction. One normal special direction and the two special directions are found. Okay? This is supposed to be out. This is supposed to be. This is claimed to be the surface of things that separate causal relations. So inside can never be described as but outside can. That's the claim. This is obviously the question. Why? Because suppose the claim is that this point is positively disconnected from the scribe plus, but this point is positively connected to the scribe plus. Well, this point is positively connected to this point. Let's say I've got a long line trade-off. A lot sometimes. These two points are positively connected. So, a signal can go from here to here and then to scribe plus. If a signal can go from here to scribe plus. If a signal can go from here to scribe plus. It can also go from here to scribe plus. So we've reached the conclusion. It's impossible that this surface that divides points that can and cannot communicate with scribe plus has a space like has a space like? No. Similarly, I want to now show that it's impossible that the surface that divides points that can and cannot communicate with scribe plus has a time like? Because suppose it was true. Suppose we have some situation here. So suppose we have, you know, something meteorized here on the surface. And I go to coordinates where I make this look flat. Which I understand. We make up a normal vector for time management. All the vectors are going to this space. This point is supposed to be able to communicate with scribe plus. Though these points can locally we can draw such a signal. So any curve that's obeys causality. All the curves that are obeying causality from this point eventually end up somewhere in this region. Any curve within the local little local light corner of this point eventually ends up in this region. So if these points cannot communicate with scribe plus causality. Because, you know, as an eventualizer this globally is a very complicated surface. But locally it's very simple. If you buy this, causality is even better. So any event that emerges from here and goes in a time like a light light project eventually ends up here. Why is it true that none of these points can communicate with scribe plus? Try to look at this point therefore. And therefore it's a contradiction. It's impossible that this is outside the event horizon. Why is this? Is this here? So neither of these possibilities both these possibilities raise a contradiction. So what possibility do we have? Namely that the event horizon is null. So it must be that the event horizon is null. Let's just say that if the event horizon is null. And neither of the two equations that we looked at occur. Okay? Of course the event horizon is null so that this 45 degree angle is null. The first contradiction was that suppose this was inside and this was outside. Then we could draw some causally connected line from here to the other side and get out. So if this 45 degree curve so the first half goes away. Similarly, the second argument was that every curve emerging from here ends up inside here. But if it's 45 degrees it's not true. The objections that we raised don't apply if the manifold is null. It's the only possibility left. So the event horizon isn't null. Remember what that means? What that means is that it's normal is null. It's easy to convince yourself that in any non-manifold there is a unique null. Suppose in this local coordinate system that we talked about before there's a unique combination of these three that are null, namely this one. Anything else would be specific. So for any non-manifold there are no time like that. And a unique null. What does this mean more physically? More physically it means that if you shine at a real light and you want to propagate along with another map this ray has only one way. That is along the normal which also happens to be attached. It's a digital manifold but anyway let me just assert it and maybe it's intuitive to you. D, suppose you take you do the following. You take the normal vector and you use that as a tangent vector to make the following assertion. And we make the following assertion which I've not really tried to prove in the moment. We are just because we're in the last lecture I'm not going to say if we lose out these sessions in December we go through all this. So suppose you take the Kevin. On a null manifold there is an any null manifold that any null manifold can be foliate by okay so suppose we're in a three-dimensional of a gamma-threading any null manifold can be foliated by two parameters that was null gld6 such that the tangents to this gld6 at every point go itself up to a normalization to the normalization. So how do we add there are all lengths 0 There are all lengths 0 So how do we actually go from one point to another? You can go from one point to another universe of lengths. Length is a formula just we choose to call them Let's say that there is a two-parameter set of null gld6 which completely well I don't know the right word it's completely that the manifold is the union of a two-parameter set of null gld6 then we draw that where there is such that tangential of the gld6 at every point up to normalization coincides with the normalization Both these universes are the same No, no, wait Let's not bring this up What I mean is that there is a two-parameter set of gld6 and looking at null manifolds that are co-dimension one and three-plus one dimensions So the null manifold itself is two plus zero two plus zero okay so there what I'm claiming is this that you can take another manifold okay you can think of it as the as so this null manifold is a three-dimension out of any ray you cover one dimension okay a parameter set of such a ray which completely constitute the null manifold the null manifold passes through one of these one of these gld6 one of these guys now these two-parameter sets are age gld6 null gld6 okay and in particular if that is true it's obvious but anyway I'm saying it's obvious because there is a unique null vector at any point that tangentials coincide with all the left and right actually I don't need to say that it's obvious because the tangent vector to null gld6 is null since these gld6 lie on the manifold at that point it's a null vector on the manifold and therefore it must be proportional to the null is the claim here now this formation is unique is unique you see because it's a gld6 it's obvious take any point solve the gld6 equation through that you know light ray starts along the manifold solve the gld6 equation no no it's a unique null vector that points over the null vector this is not true for normal null variables what? this is not true they cannot be non-unique for you sure sure sure we give a physical interpretation something like what we say for null variables okay so look on the null manifold it's a unique null vector at any given point but the tangent to the gld6 can only be that point and the gld6 equation for a null the null gld6 equation is completely characterized by extension two parameters because this runs through one dimensional line now we start with another point forget about the gld6 but two plus one is three so to make up the whole null manifold so gld6 so all the points one of my null gld6 will be described by the same set set of parameters set of parameters that describe the gld6 of course I was going to be telling you about you can try to parameterize the manifold by affine parameter long gld6 and with gld6 so if you want to parameterize that manifold just check this is the one single parameter we are telling you that it was m squared or something that it was zero yes it's not zero yes we have two directions you see suppose it was not zero suppose we had a manifold in time meaning by which I mean a space right now time is part of the okay the span of this manifold by gld6 what you can do is imagine that you have a dust particle for this manifold you choose an initial surface and you keep the arbitrary velocities of the instrument many different ways in which you can do that all of which will give you gld6 that make up the manifold okay you have to lie firstly it's not even clear that you see there are many things about it firstly in arbitrary sub-manifold there won't even exist gld6 that will stay on the manifold in arbitrary manifold gld6 will leave the manifold there are various things about it so it's not even clear that it's possible to orient it okay when it's possible it's not clear that it's unique do that the number of parameters yes this is a very general thing that applies to two dimensions okay it's a 10 dimensional space you would have a 9 dimensional eventurized okay but you have a 8 parameter set of countries in the context of the eventurized what are gld6 these gld6 are the rays of light that are trying to make it out to scratch the but never quite making it you see outside the eventurized and go to scratch this inside the eventurized it's nothing that goes to scratch this all the light rays fall into the city light but on the eventurized there are these light rays that are at the edge they don't quite make it to scratch this neither do they fall into the city light they just stay suspended in particular in our particular these light rays are staying here so these light rays so these are the light rays they are vanquishing you try to shine your torch out to it because as far out as it can which constitutes neither going out nor away and these are the gld6 that make up your this making the eventurized is the gld6 that we want not parallel in the sense that they can expand area crystal but yes it's a good point that if your space time is I am going to describe that okay is this clear any questions about so if a dust particle or something penetrates the event horizon and that light which is suspended at event horizon gets scattered by it it can leave it cannot but there is another another direction suppose you have a day of life it's going global and then you somehow scatter it it can go inwards but it cannot go out you see all scattering happens within your light and all the eventurized scattered here are not out because another thing prevents you from it very hot excellent now I should say that these light rays are no fictional and nobody is claiming that okay what if some matter falls into an r equal to m increases whether the manner falls and r equals to m increases then that tells you that in such a space time r equals to m is not quite there it's a very good question so the question was without ascending into can you take this as the last problem here your problem says consider ds squared equal to minus d squared into 1 minus m to v of by r plus 2 dv this is the very diametric that describes back hole formation due to things falling there is a kind of thing that we have in mind because m is a function of v okay because m is a function of v it's changing now if m was a constant the event horizon manifold the event horizon manifold would just be would just be precisely m is equal to 2m is equal to r is equal to 2 okay the exercise is very but slowly so that m dot is smaller compared to whatever should be smaller bigger m squared let's think more carefully m is an L so I want to dimension this condition m dot maybe it's just 1 but if you need to assume that small compared to something else m dot is very small okay compute the event horizon of this okay given this and given also that m dot as v goes to infinity and it tends to this faster than anything that you let's say that the technique is exponential at large times it approaches m dot very fast given these two things compute the event horizons in a power series expansion this is an answer question okay all you need to do in order to do this is to remember that the event horizon is a null answer but just in order to answer this question try to write down the manifold manifold is described by some s of r r is equal to z because there is no angle to the problem it's going to be null and it has to at large late times tend to r equals to m these two things completely define the manifold it's not a difficult exercise it's an interesting exercise so what you will find is that at time v the leading order in your derivative expansion at any given time v r will be equal to m of v but there will be connections to this suppressed by derivatives find the first number should we take as a definition the non-standard space to remember that so null manifold that separates curves and communicates with scratch class from those that they don't remember a space line scratch class is well defined some points can communicate with scratch class some points can it's a purely grammatical thing can you draw a causal curve that goes from this point to scratch class if you can it's insane it's not hard we have non-manifold for a lot I mean that problem is when you lack of some for one direction at some point how would you know what happens to every s so the investorized manifold becomes single it's always true what happens this curve is that the event horizon topology changes in one case we have this condition there will be a zero in every case there will be two disconnections and when you bring them up it will certainly become one look at this for instance suppose you look at the following equation suppose you look at the equation x square minus y square is equal to alpha let's plot these curves when alpha is positive okay so y is equal to zero x this is alpha positive let's plot the curve when alpha is negative symmetric like this what happens when alpha is zero it's x plus y and x minus y it becomes like this and alpha positive alpha negative we've got smooth curves it's a manifold it's not a manifold so this is what's going to happen okay you have two event horizons there's some point of touch in some singular way you know on a large amount there's no right way to do that yes at that point at the time that it touches it's not a manifold it's a manifold by definition every locally patches isomorphic to a manifold at that point it's not so you can use these manifold notions just before and just after this is a singular singular thing to the event horizon no claim to singularity in physics just this concept auxiliary concept that you define it values the singularity okay I want to tell you more about more about more about these okay and okay just so that we finish what we're talking about now I'm going to prove things very lightly beautiful discussion in terms of chapter six of these so for those of you that I strongly recommend this class is going to be chapter six the thing I'm going to try to I'm going to tell you about is one of the one of the classic and most important terms of general relativity namely Hawking's area at least there okay so to do this we're going to start the collection of geodesics that foliate a manifold it's called a concoct it's just a name so collection that foliates a manifold use the word concoct so what we have to do is that any null manifold admits a null concoct null geodesic the congruence of current geodesics okay now what we're going to try to do is to understand how these how these congruences evolve in time evolve in more precisely affine parameter a lot of these things but I'll never try to tell you precisely okay so now let's look at any given value and any given value has its tangent vector I'll call it B okay this tangent vector in some local coordinates we'll look at 1, 1, 0 so what we're going to do is suppose we had a congruence of time like curves in particular curve there's a three-dimensional there's a there's a distinguished three-dimensional subspace spatial subspace there's a spatial subspace that's to the tangent that we want however suppose you've got a null congruence okay or just a null geodesic there's a null geodesic and I say let me look at the orthogonal subspace okay the space of vectors that is orthogonal to this this geodesic at any point this of course as we've seen includes includes the geodesic itself again a three-dimensional space but includes the geodesic itself because the geodesic is null now what I want to do is somehow define a spatial direction that is orthogonal to this subspace the kind of thing that I want to do is to focus on these two directions I want to define I want to focus on the x-y directions for what reasons it's okay so okay yeah so what I want to do is to focus on the x-y directions of this okay so in order to do that how could I do that but one way of doing this is the problem look the vector of one this one is zero this is a vector number called t because another vector is going to be called n okay in the square I have this spatial vector one minus one now let's look at the algebraic relations okay pi and t firstly of course t squared is 0 t is okay I'm going to put t dot n okay t dot n is right so t dot n is 2 so let me put a one so t dot n is 0 so what I'm going to do is the following along my now I've got this null junction suppose I've got a vector a such that t dot n is equal to one at some point and then you take the vector and parallel transported along along the gdc parallel transport doesn't change scales we know that the tangent vector along the gdc changes by parallel transport if n also changes by parallel transport in particular t squared n squared and t dot n all don't change okay so given one such vector which I've chosen if there's any immediately on the gdc and then define like a field such vectors or one parallel set of such vectors okay along the gdc I defined it by parallel transport so my definition of ways in particular basically the condition that that t dot v of n be the same because the n view field is defined by a parallel transport that is what changes the parallel transport okay now given this p and this n okay for any gdc I define the projector or subject to t and n okay we do it but here in this case what would we do we would want e w view okay we would want minus e w view in this case negative okay and then I would want to get rid of the one zero plus so I want to do something like plus t new n new plus t new in and now we check what this works okay so this thing is minus one one point one this guy let's look at zero zero zero zero gives one no yeah this thing with a minus is minus one one and this thing zero zero with t new and new would give one one one both ways would give one one the one by square root two would give half yeah so this works okay well with the space guy that way it works so this would just become zero zero something you can check so this operator gives you a projector onto a two dimensional space like plane orthogonal to both the tangent and it's null counterpart okay it's projecting onto a two dimensional spatial another part is not any right yeah let's let's think about let's let's work around here you have to assume it doesn't assume it doesn't matter just one so of course what you could do is choose one and any normal that would be other and then by normalizing this appropriately you could get the other product right so this is not unique it's not unique and it's not going to matter what we choose what is important is that once you make one choice you then make the same choice all through and that's this question that is once you make one choice of n mu the n mu's along the geodesic are then defined by parallelism okay so that's what I'm saying so we got these you take some slice let's say some space like slice and you have to cut it on the space like slice you get to make a choice of this n one for each time it cuts you get to choose this but after that so there's ambiguity but after that how it evolves along the geodesic it's unproduced okay that's all we will need to do and then there's something okay yeah now I'm going to tell you about various things that you can very simply prove maybe the first of these are true for you and then a very small algebraic exercise is going to do that are very simple to prove they're all very important, very detailed in terms of geodes I'll prove one again for you so one thing is at the end of these algebraic exercises what is it? okay so let me start so the first thing that you can prove is suppose p times beta is usually at any one point so it's losing a little vector in the tangent space in the one point around any geodesic this vector lies in your orthogonal direction so this projector is your orthogonal projector okay so suppose p times eta is equal to eta that is you've got a vector that is in your two dimensional set special sets whatever you're finding think I'm going to try to prove with the following that if p times eta is equal to eta that p at one point lies with inverse then you have to follow the equation d eta mu where d means derivative of the tangent d mu is equal to p mu lambda p lambda rho p and where p is is equal to at times eta theta the index of the equation is mu is equal to no lambda rho rho nu I'm going to try to prove it for you in the rest of the lecture I will prove it right out of point let's prove this one it's the most important just to show you how easy it is okay but what is this equation saying this equation is saying that evolution of eta along the geodesic it's given by some linear transformation on eta okay so that if you try to you try to parallel to transport this this vector eta it's it rotates in some way that depends on eta but the important point of course this model that I talked about anyway the important point is that you got these projectors so the change in eta is always in the projected actions so what I mean is that suppose you take some eta that lies in this two-dimensional subspace orthogonal to both these two-dimensional okay then you take this eta and transport it by parallel transport along the geodesic it remains orthogonal to this null subspace more or less obvious but it remains orthogonal and remains orthogonal by this rule then you get a particular rule for how this thing moves along okay let me try to show you this this process this way let's start by t dot d of eta this is very important this is equal to t dot d of p mu mu eta mu t dot d on n and p dot d of t so p mu nu so what is eta what is p mu nu? p mu nu is eta mu nu okay plus n mu t nu plus d of t nu but t dot t of both m and t t dot d of t is equal to d of t is a geodesic question and t dot d of n is equal to 0 this is not a final equation of this feeling so as far as this operation t dot d is concerned both t mu and n mu amongst the vectors so so so we can you know add this p mu nu basically for free because you can just pull the vectors outside okay let's write this out this is t dot d of eta mu nu eta mu plus n mu t nu acting on eta mu give 0 of this you just take this on this side and that's 0 this is just because p eta is equal to eta is equal to p eta this is a triviality sorry sorry but now let's see so the next step is we will write this as this is equal to now I can bring the p mu nu out because t dot del commutes to p mu so I can write this as p mu nu times t dot del on eta mu okay we can do this okay and now we have this definition of v when we had v nu rho is equal to what was it del del rho so now you write that using that sorry we write v alpha v is equal to v alpha one more thing that we need here that is that is that the other thing that I didn't tell you about is the following look see suppose I have my suppose I have my components okay suppose I have my components here and I choose coordinates for my null manifold to be a coordinate which is which coordinates and a coordinate which is wound up which tells you which goes around this affect how we develop the genesis okay now my these null coordinates is n mu okay can be thought of as pointing from one one genesis to the next one you see these n mu yeah so that my null manifold okay remember that my null manifold was basically spanned by this at any given point there was this null direction and then there was space like direction okay so if I wanted to point from one guy to the next guy okay this is some linear combination of a space like vector and this null vector okay and this condition that I am choosing let me say suppose I have some arbitrary y's okay if I draw little vectors from one genesis to the next these little vectors are not unique if I could draw this guy or this guy okay but the non-uniqueness is by is by shifts in t shifts in t remember t has minus one of this a vector basically suppose I suppose I take a little vector pointing from one genesis to the next a little vector pointing from one genesis to the next okay and I choose this vector to belong to this orthogonal subspace the orthogonal subspace that we were talking about that picks up a unique direction okay because we shifted by anything proportional to t that would spoil the orthogonal so it would not have to be orthogonal to and it would continue to be orthogonal to t but not to n so it would not have to be orthogonal to this whole, it would belong to this whole protected subspace okay good so you can look at little vectors you can look at these n vectors as being a choice of vectors that go from from one genesis to the next okay and use this to define use these vectors to define the zeros of lambda okay you know you can you can find parameters along each of these genesis now these applied parameters can be shifted by anything independently genesis equation okay now what I tell you is to take these little orthogonal vectors that are normal to n okay and define that if lambda is equal to zero here lambda will be zero along along the line everywhere okay so that gives you like that gives you a surface of constant lambda this defines your surface of constant lambda now with this definition this is the surface of constant lambda clearly these vectors these n vectors are the vectors del by del yf at constant lambda or the proportional to that because you're going from one genesis to the other but at constant lambda okay so these n vectors can be thought of as proportional to del by del yf and these these these stuff we thought of as proportional to del by del yf okay but now del by del yf at constant lambda commutes with del by del del by del lambda at constant yf but del by del yf sorry lambda at constant yf is the tangent vector so these vectors here can be chosen to commute with the tangent vector and that is actually what we need for this so let me you see suppose yeah so suppose but the eta vectors I mean del by del yf okay I mean del by del yf then we have that these vectors commute and therefore g mu g mu on okay let me sorry I should have okay let me show you this okay suppose I'm sorry about this so t is this vector that we've talked about before and del eta is the vector del by del yf at constant yf and these vectors commute d commutes with both eta okay so that tells you that d mu of actual displacement on this sorry d nu d nu of eta mu alpha minus eta mu alpha del mu of d mu is equal to 0 by definition of the vector's commute okay now because this is true it's also true that del nu of eta mu alpha minus eta mu nu alpha del nu t mu is equal to 0 and the reason that this goes to this is that the two terms cancel because of the energy you see the way they differ is by a term that is d nu gamma mu nu theta eta alpha and this one is eta mu alpha gamma mu nu theta d nu theta but these are the same okay the point is that this the only nonzero index is here okay so these two are now the same but this quantity is what we call d okay and this quantity is what we had d dot d times eta is what we had here okay so you can replace this by an eta dot b and that gives us this okay sorry about that sorry about that so let me let me let me let me let me summarize again okay so the eta's for which we are doing this are the del by del y of this okay because these eta's commute with rectity this goes over to to this and we've got this equation and once we've got this equation we can rewrite this as p b p eta because p has it okay so we prove this equation that we want sorry about that what is important here is the fact that t n get and eta commute t n get eta commute I'm trying to find a surface I'm trying to find how these normal vectors evolve as we move and the normal vectors define these normal vectors define the sort of coordinates of that surface I'm trying to see how the surface that's evolving how it shares compresses and moves okay now we've just got like a few minutes left let me let me very quickly so from this equation okay we've got this important quantity b and b here so b is an effectively 2 cross 2 matrix because it's orthogonal to the other directions it's a matrix it's really a 4 cross 4 matrix but it's projected on both sides so it's effectively like a 2 cross 2 matrix okay and there's some standard stuff you can do you can define theta is equal to this b hat mu so b hat mu v v v okay so theta is equal to b hat mu mu sigma hat mu nu is equal to b hat mu mu select rise minus the trace point and omega hat mu nu is equal to b hat mu so this is the answer so in terms of this of the of these components b hat can be written as is equal to half theta mu nu plus sigma hat mu nu plus omega hat so what I've done is I've taken a probability 2 cross 2 matrix and write it as symmetric traces plus anti-symmetric plus trace matrix okay so it's just that you can do my matrix okay now one thing you can prove we'll find this now to the null GeoD6 that go around the event matrix okay we haven't yet here used that this congruence of null GeoD6 is the congruence that is normal to a null matrix that's just any bunch of matrix okay it's normal to that this congruence is the congruence of the normals of another manifold allows you to show and I will try to do it now that omega hat now you define the following thing in this little congruence of null GeoD6 you define the following area okay you define a is equal to epsilon mu nu epsilon with the square root okay so epsilon mu mu rho sigma t mu eta mu that's n n mu and then eta 1 rho eta 2 what we're doing we're taking two little two little vectors in this normal space it doesn't matter which ones so two little vectors in this normal space these vectors span an area just a two-dimensional area okay and just the usual standard definition of this two-dimensional area is written as follows basically you know projecting on the useless one zero and one one and one minus one direction and computing just the area of the spatial two-dimensional space so you so suppose you've got this congruence of GeoD6 and between any so we've got this two parameters out of GeoD6 we've got another GeoD6 here and another GeoD6 here and any given time given outside parameter you've got these GeoD6 have a set of two-dimensional associated with them in the sense of this form okay now another thing you can quite easily show is that dA by dT is equal to T why is the why is it you know areas always how do you define the area of a triangle in this sense yeah it's a one eight to minus eight together so whenever you try to measure a volume it's measured by it's actually by a form the duality you know you dualise it to a scalar that's this epsilon and then measure that scale you also associate okay so something else you can prove I'm sorry I'm not going to prove all these things but it's very nicely done in this section okay is that dA by dT is equal to T so what's the picture with you suppose you've got an event horizon surface and you cut the event horizon by some spatial two-dimensional spatial map you can associate an area with this an actual horizon a few bits of area between each of the two between neighboring geodesics now as these geodesics move around these area elements change okay what's happening is it's like a fluid as the geodesics move around you can think of the geodesics following the fluid particles okay so the fluid particles may move away from each other or move towards each other is this the same thing the fluid particles can also shear that is you know move like this and do funny things without changing the area that shear is associated with that shear is this thing and the fluid particles can also have some vorticity okay and vorticity is this over here for the case of non-manifold there are only vortices here so it's vorticity free flow and vorticity free particles are moving so as to generate essentially vorticity free flow with just shear and compression okay great so all this is very nice but the most important thing comes is what the equations that are of the salt that we describe that again describe very nicely in these lecture notes right you're doing form the boundary equation you found an equation for the derivative as you move along as you move along okay if I'm the equation the derivative equation then d d theta by d lambda is minus theta squared by 2 minus sigma mu mu sigma mu plus omega mu mu omega mu mu minus or mu mu d mu t and basically all you you start by d theta in order to derive this you start by d theta by d lambda d dot l is just a definition of d mu mu sigma theta is the trace of speed and then you will use various manipulations you do various manipulations all the time using the using the gudc equation and the fact that the ends are transported there with a few random manipulations you can start with this equation and reach this beautiful you have to note at the bottom this thing here is negative definite this thing here is negative definite you might wonder is it really negative definite because both its squares and relativity can be either positive or negative depending on whether this makes like a time limit but theta of course is the second one and sigma is the tensor completely in the space-life direction so this kind of is negative definite this guy here is negative definite this guy here is positive but its just zero for the case of interest that is not there and then you also have this this term the thing that is multiplying r mu mu in this term is is an algebraic let me put that into vectors along these genesis so now lets use Einstein equations Einstein equations relate r mu mu to T mu mu and to T times G mu mu but G mu mu the G mu mu part vanishes because the T's are not T G G is there so you can basically take this and replace this by T mu mu with the right hand


Hey guys, welcome back to Tech Talk. My name is Chase, and today we're going to be discussing how to properly clean your SeaKeeper Gyro. Alright guys, so today we're out on the Open 282. It's equipped with the SeaKeeper 1 Gyro, but this is something that we really want to focus on, especially on the SeaKeeper 2 and the SeaKeeper 3 that are not in a housing. So you can actually get to the inside and hit some of those internal components. But all we're going to be doing today is just giving a light rinse. So it's kind of going over the unit. As you can see I got my hose in a nice wide pan. I'm not spraying anything directly. I don't want to hit any of the electrical components directly. Just a nice light rinse. You can think of this as flushing your motor. Obviously running your outboard hard all day in salt water. It's always a good idea to flush it and get that fresh water through all the internal components. This is the same principle. And it's just as easy as that guys. Alright guys, thanks for watching this short video on how to clean your SeaKeeper Gyro. If you have any questions or comments, leave them below. Like and subscribe. And if you have anything that you guys would like to see from us, please email us at the address listed below. We'll see you next time.


Okay, so this is a gratitude video. My best dates of 2022, and I'm throwing the challenge to everybody whose name I'm gonna mention in this video right now, that you should also look back at 2022 if you can, if you want to, if you will, and tell us which of the dates in 2022 if you can remember, or incidents. If you don't think that exact dates are going to happen, maybe a time around a certain month that certain things happen for you, you can do a video like that. So that's a challenge I'm throwing to all YouTubers who are watching me right now, especially me and AJ, me and M.O. If you want to do a separate video yourself, Nana Shanti, Anjyua, yes, I'm supposed to record this video back in Ghana and do a giveaway, but for some reason, well, adulting happens and I couldn't do it. So let's play Emotion controller, headless YouTuber, Dakmak Mapo. All of you guys, if you're watching this, you can take up the template and do a video. I would like to hear how your 2022 went and what you're looking forward to for 2023. So yeah, my year 2022 started off with me wanting to do more quality work, work that tested my patience, work that required that I work with other people. And this is something that I think I professed silently or put it out in the universe silently. And I think the universe took it very personally because I told myself that I hope to not do too much of client work in terms of documentaries and other stuff and maybe just concentrate on voiceovers and my own kind of work, the work that I want to do, creative work, films and other exploits. And the universe took it up. So the beginning of the year, I wrote my first short film, which is Jolt. Jolt has been an incredible experience for me in 2022 in filming it, in preparing, in meeting the people that I worked with, in collaborating with other creatives and putting it out in festivals. If you haven't seen Jolt, I'll link it for you to check it out after this video. So I finished writing Jolt in January. That the first January is a date to be precise. That's when I finished writing Jolt and I sent it to the producer of Jolt, my producer, Mauly, who we had been talking about these projects in December, 2021. So I told him that we need to work on it and then we did. I finished writing Jolt 31st January. So that's my first achievement or best date of 2022 starting in January. My next best date was 18th March. It's the dates when Elaine and I actually decided to start trying for a baby. And yeah, so that's when I remember that we had the conversation and took it upon ourselves to roll it out or start an action plan to start trying. So 18th of March is also really important to me because in the end, something came of it and it's been beautiful so far. Second April is also another amazing day for us. It's the day we went to pick up our baby girl, Nala. In addition to the family where we did the vlog for and everybody was expecting it to be a baby so we didn't also know that the baby was on the way. So all good, Nala came into our lives on 2nd of April, 2022. And yeah, it's been beautiful ever since. So 2nd April is one of my favorite days as well in 2022. And on 12th April, just barely two weeks after Nala came, Elaine's friends visited. We went on a group trip to Boussia and it was such a beautiful getaway for a group of people or a group of friends. It was, we went as well with Nala. So it was the first time traveling with Nala. We were just living it or just doing, I mean, as we would. And I really enjoyed that Easter holiday as well. So it's one of my best dates. The travel was smooth, Nala was a champ because we traveled for about seven hours to Boussia and not at one point did she give us any trouble or try to, you know, she didn't even puke. So 12th of April and yeah, from there, 16th of May we, no, actually no, hold up. 16th of May is my next date and I got accepted into the AU Media Fellowship. Yes, I got accepted into the AU Media Fellowship on the 16th of May. So the story about the AU Media Fellowship is actually funny because I had almost missed out on applying literally 24 hours to the deadline. Elaine and I or Elaine said that no, I should do it. She had told me way before but I was dragging my feet, not trying to do it. And then a friend of mine who is part of the program sent me a message asking if I'd be interested in wanting to join and 24 hours to time or to the deadline, Elaine and I put together mostly Elaine put together the application. Well, I had everything prepared but she did the typing, most of the typing because I told her if you want me to do it. 11 days later, still in the same month of May, we found out that we were going to be parents. I think Elaine already knew, we've already told you the story, she felt that she was but she needed to tell me in person. So on the 27th of May at the airport is when we actually talked about it for the first time with the famous, I've not been truthful to you. I don't remember exactly the quote. What's the quote? What was the quote you said in the car?  Yes, I haven't been truthful with you. That was the exact quote on the 27th of May in the car. And then she told me about how she was feeling and then we tested and it was beautiful. Yes, so 27th of May, 2022 is when we found out that we were going to be parents. And on 4th of June, things were happening like really back to back. And I don't know why I kept thinking that not so much was happening because literally 4th of June was the day we shot Jolt. And it was a full day. Behind the scenes, I'll link it. Is it here? Does it go here? I think it goes here. You can watch the behind the scenes video of Jolt, how we shot it in one day, but we have been prepping since I finished writing Jolt in January. Dealing with the actors, having rehearsals, trying to find a schedule that worked for all of us, all of us who are going to be part of the filming. So 4th of June is when we shot Jolt. And on the 29th of July was when we premiered it. We had worked on the edits back and forth. We edit music and everything. And this is the most work I've ever done with the most people as well, allowing people to do what they are best at. And I can never be more grateful than I am for the experience of Jolt. Everybody that was on Jolt's from filming it, editing it, sound, graphic design, everybody that made sure that Jolt's became a thing. Thank you, thank you so much. So yeah, Jolt's was premiered on 29th of July and everybody that came to the premiere, everybody that posted a poster or reshared it some way, somehow bought tickets to the premiere. You guys are the true MVPs. You really made my year 2022 an absolute beauty. And I'm truly grateful for that one. 7th of August, after that we had finished premiering, I put it out for festivals. And on 7th of August was the first selection that we got the first selection. And I think it was a Swedish film festival, Bauden International. And that was our first selection, 7th of August. On the 25th of August, of course, we had already discovered that we're pregnant from 27th of May. On the 25th of August is when we found out the gender of the baby. We're probably gonna tell you the gender at some point, but I don't know which day. And it dawned on me after we discovered the gender that in a way, whatever I wrote in Jolt, in January, was literally manifesting. Yeah, was literally manifesting. So this was one of the days that I was like, whoa, what's going on? So 27th or 25th of August. And on the 31st of August, we went on a holiday to Netherlands. I mean, we do a yearly trip usually, at least once every year. And I really enjoyed that too, because it was one of those times where... Babe, was it my first time since I became a freelancer that we came? I think so, right? Yeah, it was the first time since I became a freelancer because the COVID happened and we didn't get to travel for two years. So it was the first time since I became a freelancer. And it felt completely different compared to when I used to come here while I was working a corporate job because there was so much freedom. It felt relaxed. You were here for just being here. So I really enjoyed my time in... Netherlands in August or August, September. So 31st was when we flew out. And of course, after that, the second of October was my first time traveling to Ethiopia. It was our second session for training or training session for the AU Media Fellowship. And Ethiopia was beautiful. I really loved my experience in Addis Ababa and the weather, the people, the culture, the traditions. Everything was really, really... And that was my first time visiting any other African country. So that was also a big deal because I've lived in Ghana my whole life and I have traveled outside Ghana to the Netherlands. But even Togo safe, I've not been driven across the border to Togo. So I think that's something that might also change in 2023. Hopefully. At least see the two other places super closest to Ghana, left and right of the border, Kodivoa or Togo. So yeah, I really loved my experience in Ethiopia. And of course, third December was the day we actually did move to a new place. It was a Saturday that we packed everything. And the moving day was, oosh, tiring. But we managed to do it. And here we are. It's been an incredible year. And I'm really, really grateful for everybody who has been a part of it one way or the other. And I'm looking forward to a much better, much quality in relationships, in experiences, in everything that we all decided to do. I'm looking forward to a great 2023, a healthy 2023. And yeah, we all have a really good one. I don't wanna continue rumbling. This is the first video for 2023. My name is Kwame, and I make videos with my partner. She hates it when I say I make videos with my wife. So I make videos with my partner. And this is me plus you is us. And you're welcome to the channel if this is the first time watching. So the challenge again goes out to everybody who just watched this video all the way to this part. And I think I need to make a final announcement before I go that in the coming weeks, once every week, there's a collaboration with the Brain and Mind Medical Center bringing you mental health education and awareness videos, having conversations with professionals. So please look out for that. It's going to be snippets on this channel and full video on Asitna. So if you see the video and you watch it and you want to continue watching the full one, which you should because it's deeper in the full video, then please click on the Asitna link in the video as you see it and then go check it out. So I'll catch you in our next video. It's peace for now.


What's up, everyone? It's your girl Kenzie here from late in sports cards, and I am rip in we've got a personal here for Michael D a 2022 Bowman inception baseball. Alrighty Michael best of luck. Let's see when you get you here Bowman inception time Here we go. We've got a William burgola Billy's a Jason Dominguez Yanks Roberto compels tigers and And a yassir Mercedes for the twins Those will be the base first parallel. We've got 33 of 75 of Michael Hernandez for the Orioles like the red look on these 75 Then we've got our first auto pirates 19 of 99 is a Lonnie white jr. Lonnie white jr. Pirates Lonnie blue foil. Oh, we're not Slum we that's where Leighton started. They're from New Jersey We are in central Florida now And we've got Royals 395 399 Carter Jensen Carter Jensen Yeah, originally started in New Jersey and they moved down here the 399 Alrighty Michael that will do it for your inception. Thanks as always for hopping in we'll get these right out to you


I know I can get pretty busy and a busy schedule can often lead to very poor food choices. That's why I always keep these five essentials in stock and you should too. First of all, always have some dark bitter greens and lectin-free vegetables in your refrigerator. One of my favorite sayings is more bitter, more better. Interestingly enough, almost all long-lived societies have preferred very bitter greens in their diet, whether they're dark green leaves like arugula or Swiss chard or bok choy or radicchio, which many people call that Italian red lettuce, but is actually part of the chicory family. There are some great varieties of broccoli that are extremely bitter, like broccoli rabe or broccoli rabe. You can find it usually right next to the broccoli in most grocery stores now. More bitter, more better. The bitterness is a sign that there are lots of polyphenols in these products. Get used to that bitterness. Seek it out. The bitterness is what you're looking for. Also, one of my favorite sayings is if you eat dark green, you will become lean. Why? Because these polyphenols will uncouple your mitochondria. And we now know that the polyphenols in these dark bitter vegetables feed friendly bacteria, that polyphenols are some of the favorite prebiotics that bacteria need. And the bacteria in turn will turn these polyphenols into absorbable compounds that will uncouple your mitochondria. So it's always a great idea to keep these dark bitter vegetables in your refrigerator. And the great news is a number of these, if you buy them whole and please buy them whole, will keep for very long time at cool temperatures. In fact, one of the reasons that traditional cultures eat cabbages, which are also quite bitter, is the cabbages will store literally forever through an entire winter. So think of storageable leaves like cabbages, like radicchio, like bok choy. Those are the ones that'll keep in your refrigerator and they'll be ready at a moment's notice when you come home and need something to eat. Second is certain nuts should always be available to you in your pantry. Nuts are a great way to get healthy fats and to help you feel full. Now pistachios are one of my favorite and pistachios are the highest source of melatonin in a food that you can get. And melatonin is not the sleep hormone. People who've read Unlocking the Keto Code know that melatonin is actually the essential antioxidant in your mitochondria that protect your mitochondria from damage. Melatonin is an incredibly important protector of your brain mitochondria. So have some pistachios. Macadamia nuts. Now be careful with macadamia nuts. They're mostly mono-unsaturated fat. They also have a very cool omega-7 fat that's really good for blood vessel health and omega-5 fat that's good for blood vessel health. But if you're trying to maintain or lose weight, watch out for macadamia nuts. Walnuts. Walnuts have a great source of a short chain omega-3 fat called alpha-linolynic acid that is one of the best health promoters for uncoupling mitochondria that I've ever found. Everybody after study, particularly the PretoMed study in Spain, showed that adding about a half a cup of walnuts to your diet dramatically improved memory in people 65 years of age or older. So my mix at my offices that every patient eats while they're waiting to see me is walnuts, macadamia nuts, and pistachios. Those are the top three. But there are others. Hazelnuts are great. Peely nuts. Many people think it may be the healthiest nut in the world. Check out episode 232 with Jason Thomas, the Peely Hunter, and see what we're talking about. Now, they're great pre-party appetizers. This is the season where we're always awash in parties and there are so many temptations. The problem is you're going to succumb to those temptations if you arrive hungry and if there's a drink in your hand. So what I do to prepare, and I actually did this last night, is I have a generous handful of my Gundry Nut Mix before we head to the party. I'm satiated. The alcohol is not going to be absorbed very quickly because my stomach is full. And then I'm not that interested in what is on the appetizer platters. There are some great nut butter alternatives. As many of you know, peanut butter is not a nut. Peanut butter comes from a peanut, which is a legume. They're loaded with lectins. The vast majority of us are sensitive to the peanut lectin. Almond butter is an okay alternative but. Pistachios have a lectin in their peel. There are several companies that now make peeled almond butter and that's a go-to for many of my patients. But there are other alibi more expensive butters. Pistachio nut butter is great. Hazelnut butter is great. Peely nut butter is great. Tahini, not from a nut from sesame seeds, is great too if you're looking for kind of that similar mouth feel as peanut butter. So get yourself some new spreads. Ditch the peanut butter and please, if you're going to use almond butter, get the peeled almond butter that you can find in stores or on Amazon. Avocados.  You should have an avocado every day. Studies have shown that people who eat avocado every day lose weight compared to people who don't eat avocados. Studies have shown that avocados in a salad or mixed in with vegetables make you absorb more of the vitamins and nutrients and polyphenols in those food than if you didn't have the avocado. It's loaded with monounsaturated healthy fats like oleic acid, which is the major fat in olive oil, loaded with antioxidants, vitamin C, vitamin K, and the B vitamins. There's more potassium in an avocado than a banana.  Should you use avocado oil and ditch the avocados? Well, while avocado oil is a good high smoke cooking oil, it doesn't have near the polyphenol content of olive oil.  Guacamole, have some guacamole in the refrigerator. There are a number of companies that now make guacamole without tomatoes, which is not supposed to be in guacamole, or use those avocados when they're starting to get really ripe and make your own guacamole. But use like Belgian endive as your dipping chip for guacamole and you'll get a win-win. All right, MCT oil, it ought to be in all of our pantries. MCT oil, as you know, is medium chain triglyceride oil. You're going to get all the benefits of a ketogenic diet by using MCT oil without the hassle because MCT oil is rapidly absorbed through the wall of your gut, goes directly to your liver, and makes ketones regardless of what else you're eating. So you don't have to go through the trouble of eating a very high fat diet or a low carbohydrate diet or even doing long-term fasting to get the benefits of MCT oil. You should also have extra virgin organic olive oil and please buy it in small glass containers because once you open olive oil, it will start to go bad and you should look to use up this olive oil very quickly.  Sesame seed oil.  We now have it available at Gundry MD. It's sourced from our same providers in Morocco that our olive oil is. It's another great addition to your diet and put it in salad dressings, pour it on vegetables. It'll add a great taste and it's loaded with fighters against these LPSs that get into your bloodstream and cause high blood pressure. Finally, dark chocolate. Have a collection of greater than 72% dark chocolate in your cabinet. What is loaded with polyphenols? Again, the higher the cacao content, the better it is for you. Have yourself one square a day. It's perfect for dessert, whether you want it or not. Look for stevia sweetened chocolate like Lily's. Costco now carries a big bag of stevia sweetened chocolate chips by a company called Bake Believe. There are on the market alieolos sweetened chocolate. I'll mention one. I have no relationship with Gatsby chocolate is sweetened with alieolos. You can find it on Amazon. Check out my most recent episode all about the amazing benefits of chocolate to learn more. Now, recently there's a new study making the rounds and this comes around every few years that there are levels of lead and levels of cadmium in chocolate. Somehow this is exciting new news. Folks, we've known that there is lead and cadmium in chocolate ever since these levels have been measured. Do not let this fear mongering dissuade you from the benefits that you will derive by eating dark chocolate. Study after study after study of people who have dark chocolate cacao in their diets show improved brain health, improved heart health, improved vessel health. So please don't don't see this latest news about lead and cadmium in chocolate dissuade you from having chocolate in your kitchen and enjoying it every night. Fermentation actually refers to the fact that most lower life forms like yeast and bacteria have to make energy by a process called glycolysis and we won't take you back to high school biology. But essentially they take sugar molecules which are a source of carbon and through a process called glycolysis they make ATP, the energy currency. Glycolysis is actually a very inefficient process for making ATP and you and I don't use glycolysis. We use mitochondria and the electron transport chain. But glycolysis is basically fermentation and fermentation takes sugar and turns it into energy for the bacteria or the fungus or the mold or the yeast and what's left over is actually what are called postbiotics. And just to review, probiotics are friendly bacteria. Prebiotics are the foods that probiotics ferment during the process of fermentation. You produce compounds that are called postbiotics and if we had to do it all over again we wouldn't use these names but sorry about that we're stuck with it. So vinegar for instance is a product of fermentation. For that matter wine is a product of fermentation. For that matter beer is a product of fermentation. And what's fascinating is almost all ancient cultures preserved their foods or made their foods edible via the process of fermentation. So fermentation of foods have been practiced actually for past recorded history. There's evidence that the Egyptians were fermenting foods 10,000 years ago really around the time beginning of agriculture. One of the reasons we think that fermentation happened was number one there was no storage system, there wasn't refrigeration, there was no way to actually keep food fresh. And in the process of just storing food fermentation was a natural process that would start regardless of what you expected to do. What's interesting is many many many cultures actively used fermentation to make their food probably far more edible and interestingly enough as many of you know fermentation in general breaks down lectins. Believe it or not a lot of bacteria love to eat lectins they think they're delicious and they will eat the lectins for instance in quinoa. Fun fact the Incas always fermented quinoa before they ate it. They let it rot and then they cooked it. Now unfortunately that's not on the package directions. In memorial beans were always soaked for a considerable period of time and most people if they bothered to look notice that if you soak your beans foam will appear on the top of the water and that foam is actually the carbon dioxide produced by the native bacteria on beans fermenting the bean. And so traditionally beans were diminished when their lectin content because they were soaked for a considerable period of time and onward and onward. So not only does fermentation really reduce lectins but fermentation gives you post biotics. And it's the miracle of these post biotics that I've mentioned in my last two books but now in gut check which will be out in January you really get to see how important these post biotics that have occurred from fermentation really are. Among the main things that fermentation does one of the main post biotics is short chain fatty acids. Now many of you have heard me talk about butyrate. One of the most important short chain fatty acids there is. And as you've heard about in my previous books but you're going to see even more in gut check. Butyrate is really the single most important short chain fatty acid for your health that you can imagine. Now we hopefully have bacteria that can make butyrate out of fiber. But it gets much more complicated than that. And let me give you a study that I mentioned before and I mentioned it in Unlocking the Keto Code. Husband and wife team of researchers at Stanford, the Sonnenbergs, did a fantastic study with human volunteers that was published last year. We've known for a very long time that fiber is really good for you, right? And fiber is prebiotics. And we've known for a very long time that our gut bacteria really, really, really like prebiotic fiber. And the thought was that our gut bacteria can take prebiotic fiber and make all these wonderful postbiotics like butyrate. And if they do this everything gets better and our immune system gets better, et cetera, et cetera. Not so fast, say the Sonnenbergs. So they took human volunteers, half got a lot of soluble fiber. The other half got the same soluble fiber, but they were given fermented foods, primarily in the form of yogurt or kombucha or kefir. And then they watched what happened to their gut microbiome diversity. Diversity means lots of different species. And as you'll learn in gut check, the more diverse your microbiome, the more different members of that community, the better your health. And they looked at the immune system, they looked at inflammation markers. Lo and behold, what was shocking was the folks who got the fiber alone actually didn't improve their microbiome diversity, nor did it have much effect on the immune system, on inflammation. But the group that got the fiber with the fermented foods had much improved gut microbiome diversity and the inflammatory markers went down. So what's up with that? Well, it turns out in my research, yes, you want bacteria to be able to make butyrate. But what's fascinating is many of the butyrate-producing bacteria actually have to have precursors for butyrate to manufacture butyrate, basically assembly line. And so one of the shocking things is that they need, for the most part, other short chain fatty acids to do the conversion. So they need, like for instance, acetic acid or propionic acid, acetate or propionate to make butyrate. And yeah, they need the fiber, but they got to have the precursors to assemble all the parts to make butyrate. And that explains why it was only the group that got the fermented foods that had these short chain fatty acids that then, when they got the fiber, they can complete the manufacturing process. It's also amazing in my studies of the longest living cultures of the world that almost all of these cultures have fermented foods in their diet. And some of them might surprise you. So let's set the record straight on no-go fermented foods. Yogurt. Now, yogurt is a fermented food. The problem with American yogurt in general is that most of our cows make a protein called KCNA1, which by itself can be a lectin-like compound, a very inflammatory compound. Now, cows in France and Italy, goats and sheep and water buffalo have KCNA2. So what I want you to do is try to avoid American cow milk yogurt. There's plenty of goat yogurt available. There's sheep yogurt available. There's coconut yogurt available. There's even peely nut yogurt available called lava. All of these are great options. The other problem with most yogurts in America is they are loaded with sugar. So you've got to be a wise consumer and actually look for the amount of added sugar to the yogurt. So look for the unsweetened variety. And if you want them sweeter, just put some allulose in. And good news, allulose is a prebiotic sweetener. That feeds good gut bacteria. Now, the second non-starter is most kombucha. Now, don't get me wrong. Kombucha is a phenomenal fermented food that has great postbiotics. But, kombucha is tangy. And Americans in general don't like tangy things. So companies put a whole lot of sugar in so many of the popular kombuchas to cover up that tanginess. So please read the label carefully. There are a number of popular kombuchas that have as much sugar as a soda. And that's not what you're looking for. Also remember that sugar isn't going to benefit your good bacteria at all. And it will only feed bad bacteria and yeast that you really don't want. On the other hand, there are some kombuchas that are low sugar. And they are available. But you've got to read the label. And just be careful of the amount of added sugar in these products. Now, sourdough bread. Years ago when the Paradox came out, I was invited on the Dr. Haas show with one proviso. Dr. Haas, who's a friend of mine, said, you're going to have to give me a bread that people can eat. Otherwise, I'm not having you on the show. True story. So I said, well, there is no safe bread.  We have to realize that almost all breads traditionally were raised, were given heft by the addition of yeast. And the yeast fermented the sugars and also the lectins in bread. And that produced an even loaf. The Egyptians are the first credited with doing this. On the other hand, almost all commercial breads in the United States no longer use yeast to make the bread rise. It's too inefficient and it's too unpredictable. That's why you get big holes and little holes. You want even holes and that's not with yeast. So modern American bread doesn't use yeast to raise the bread. Now traditionally sourdough bread is made with a sourdough starter which uses these yeast and bacteria. But most sourdough bread in the United States, it says sourdough because they use sourdough, but most of the bread was still unfurmented. So buyer beware sourdough bread in the United States for the most part is not what you're looking for. Okay, so let's get to the good stuff because there's some easy to use good stuff. Number one best fermented food that's easy to find and use is vinegar. You've got to give the assembly line to make butyret what it needs and vinegar supplies acetic acid, the short chain fatty acid that it needs. Now if you want a deep dive about other videos all about the benefits of vinegar, go to my YouTube channel and check out vinegars. Now acetic acid is also a communication system between our gut bacteria and our mitochondria. And acetic acid as well as butyric acid are one of the best ways to uncouple mitochondria. And once you uncouple your mitochondria you get profoundly good health. Probiotics, even if they were in vinegar, are long dead. But those dead probiotics also carry information. So for instance, apple cider vinegar with the mother. The mother is actually all the dead bacteria and what's left over of the polyphenols in apples. And those dead bacteria also carry information. And it's a really exciting information that you're going to learn about in gut check. Dead bacteria tell tales. Now vinegar is a great way to do this. Vinegars make you actually uncouple your mitochondria to lose weight. And one of the miracles of weight loss with vinegars is this uncoupling effect. Now I love aged balsamic vinegar. It has multiple available polyphenols. It's actually the richest source of resveratrol that you can get. I happen to have a brand that I recommend. I have no relationship with them. They're called Napa Valley Naturals and you want the Grand Reserves. It's really thick. And you just need a teaspoon or a tablespoon to make my famous fake Coke. You use sparkling water. I prefer San Pellegrino. I have no relationship with them. And you put in some balsamic vinegar. Stir it around and presto change. Oh, you have got a health enhancing compound that tastes really good. And it tastes kind of like a fake Coke. Apple cider vinegar, champagne vinegar, red wine vinegar, sherry vinegar.  I've got probably eight different vinegars that we mix and match in our salads. And it's a really easy way to get these post biotics that you need. Now here's a shocker. My second best favorite way of getting these fermented post biotics into you is raw cheese. News flash, raw cheese. What does raw cheese mean? Well, it turns out cheeses, particularly in Europe, do not require pasteurization to make it into the United States. The bacteria and the yeast in milk actually impart great post biotics in the process of making cheese. And there's particularly some raw cheese bacteria that actually promote weight loss. And surprise, surprise, three of the blue zones that I've talked about before and you'll learn more about, you know, gotcha. Three of the blue zones eat large amounts of raw sheep and goat cheeses as part of their diet. Now there's also a incredibly interesting component in whole milk, which is called milk fat globulin membrane.   Wow. So you get a double benefit. Now here's number three that is going to shock you. And this is straight from long lived people. Most people probably don't know that the longest life expectancy in the world by country is in Andorra. Now Andorra is this little tiny country between Spain and France up in the Pyrenees Mountains.  In fact, a nearby town in France, Toulouse, basically lives on sausage. And Toulouse, despite living on sausage, has the lowest degree of heart disease in all of France, despite having this massive high saturated fat diet. What gives? It turns out that traditionally sausages are cured, are aged by the addition of bacteria. And bacteria eat the sugar molecules in the meat. And they also, fun fact, eat new 5G C, which is in beef, lamb, and pork, which, as many of you have read, is a real mischievous molecule that really harms our health. But fermentation of meat, of sausages, eliminates new 5G C. And it also eliminates new 5G C in milk products. So, strange as it may be, traditional sausage making is a fermented food that has postbiotics, fermented things that improve your health.  Now, not so fast. Most sausages in the United States are not fermented.  Most sausages with an Italian name applied to them, like pepperoni and salami, are not fermented. So buyer beware. But you can find traditionally fermented sausages that are made in France or Portugal or Italy. And they are findable. And those are actually remarkably good for you. Who knew? So, vinegar, aged raw cheeses or aged cheeses from Europe in general, and traditionally prepared sausages. One more last thing. Many of you know I have a very good friend, a three times James Beard award-winning chef, Chef Jimmy Schmidt. And Jimmy Schmidt produces sausages and bratwurst. And lo and behold, Jimmy inoculates his sausages and bratwurst with bacteria. FDA approved, I may remind you. And I was talking to him one day recently. He said, what the heck? How did you know to inoculate your sausages? And he said, are you kidding? That's the only way you could ever make a sausage legally in Europe. So I'm just continuing a tradition. So if a three times James Beard award-winning chef knows what to do, I'm passing it on to you. Eat your fermented foods and buyer beware which ones to avoid. I want to talk about three of the most misleading food labels out there and how to avoid getting tripped. First up, no added sugar. This may be one of my favorite ones. You see this on juices, dried fruits, and a lot of other snacks. What they don't want you to know is that this basically means that this product already has so much sugar in it. We didn't need to add any more.  So the other thing you want to look for, so many labels now say added sugar. And you look at added sugar as what you actually think is in that product. In fact, the system's being manipulated to disguise all the other sugar that's already there. So sure, if you see a product with a lot of added sugar, you want to be aware that that's probably not a product that you want to use. But even if you see very little added sugar, that doesn't mean you should go look for the real sugar content. So how do you find the real sugar content? First of all, don't look at total sugar. Look at total carbohydrates. That'd be one of the first things under the carbohydrate level. Right below that, you'll see fiber takes a total carbohydrate number minus the fiber. That will actually tell you the entire grams of sugar in that product. Now, what does that mean? Well, one teaspoon of table sugar has four grams of sugar. So fun thing to do is take that total carbohydrates minus the fiber. Take that number in grams of sugar divided by four. And that'll tell you the teaspoon of sugar in that serving size. Now remember, companies are very clever. They reduce the serving size to get you a number that you'll feel pretty good about. The other thing companies have done because of labeling laws is if there is a half a gram of trans fat per serving, you don't have to list trans fats on a label. So many companies will take the serving size, reduce it to just the level where they no longer have to show you trans fats on a label. But nobody, quite frankly, is going to eat the serving size because in each package, there may be two, three, four, five servings in a package. And it's incredibly unusual for human behavior to not complete the entire package. So buyer beware. Let me give you an example. We had the former head of the FDA on our podcast the last year. He uses the example of a bagel. So a bagel usually has around 330 calories. And if you look at the label, it will say zero sugar. But in fact, if you do this math, you'll find that the total carbohydrates of the bagels, about 33 grams of carbohydrates and zero fiber. So that means 33 grams divided by four. That's about eight teaspoons in that no sugar bagel. And as we talked about on other podcasts, when you finally grind up grain products, you actually make the sugar more accessible for absorption than plain table sugar. In fact, that's why bread as a glycemic index of 100 and table sugar, which is half fructose and half glucose, by the way, it's sucrose. As a glycemic index of 85, you're actually getting more sugar in from a piece of bread than sugar. Wow. And you can't even taste the sugar in the bread. So buyer beware. Now, another one of my favorites is organic free range or cage free eggs. It sounds pretty good, doesn't it? Wrong. First of all, free range and cage free chickens merely describes that the chickens don't have to be in a cage. It doesn't mean that they're allowed to go outside. Commercial chicken warehouses can contain up to 100,000 birds in a warehouse jam so close together that they actually support their own way because these chickens have such osteoporosis in their legs that fully 10 to 20% of chickens that are sold for frying chickens actually have broken legs because they couldn't support their own way. And by the way, there's some fascinating evidence that it's the lectants in the corn and soybeans that fed these chickens that are the problem. So a free range chicken by law can be kept indoors for its entire life. As long as you open the door to this warehouse for five minutes every 24 hours and the chicken has the potential to go outside to a three yard by three yard by three yard patch of dirt or grass. And by frankly, those 100,000 chickens aren't going to get through that door in five minutes. So but that's the definition of a free range chicken. Chickens are fed corn and soybeans.  And that's why so many times you'll see organic cage free eggs that are quite frankly pale yellow in the oven. They have absolutely no of the vitamins and polyphenols and beta carotines that a chicken would normally eat when eating bugs out of the field. And by the way, you can't certify that the bugs that the chicken should be eating are organic because you didn't cast all those bugs. So look for pastured chickens. If you can't find pastured chicken eggs, your second best choice is Omega 3A. Why? Because the chickens are fed either flax seeds or algae or both. And they have Omega 3 fats in their yolks. Those Omega 3 fats are alpha linolenic acid. And if you looked at my last book, I've read it, you'll notice that alpha linolenic acid, ALA, is actually the one fat that made the difference in the famous Leonhardt diet study in preventing coronary artery disease. So imagine that you're going to get a nice dose of ALA every time you have Omega 3A or an algae-based egg. And there's several good manufacturers of algae-based eggs. With that in mind, eat the yolks and give the whites to your dog. Your dog is going to love it. You're going to get all the benefit of the alpha linolenic acid. And you'll also get a healthy dose of choline for your brain simultaneously. Okay, number three, light-bored diet products. Now, almost all of these products with this label contain artificial sweeteners like saccharine, aspartame, sucrose, which are practically poison. You might know them better by their brand name sweeten low, you will explain them. The problem with these sweeteners is that they actually kill bacteria. So imagine that you're having a light or a low-fat, low-sugar yogurt and you're thinking, hey, this is great. I'm going to get all these wonderful healthy probiotics from my yogurt. But in fact, the study from Duke University in 2007 showed that one pack of Splenda sucrose can kill off 50% of your gut bacteria. So why would you eat a potentially healthy low-fat, low-sugar yogurt to get the probiotic? When in fact, the sucrose in that product is going to kill whatever you swallow and half of your gut buddies simultaneously. Talk about a bad idea. When you kill off your gut bacteria, remember, you're going to drop more and more bad bacteria into your place. Replace them. That's going to cause more leaky gut and you're going to gain weight due to the inflammation that follows leaky gut. Now, if a diet food isn't high and sweet, it's often loaded with fat and artificial flavors instead of a healthy fat. One more interesting thing that I've written about in the past, but I think it bears repeating. Anytime you have a fruit in a yogurt or if you even take one of these low calorie yogurt and you add some blueberries or you add some blackberries, you should know that that milk product will actually bind to the polyphenols in the fruit that are beneficial to you. You'll get absolutely no benefit from those polyphenols. This has been shown in multiple human studies where people are given blueberries, for instance, or blueberries mixed with yogurt. And, more than behold, if you look at their blood levels of polyphenols and those metabolites for one hour, three hours, and six hours after they ate them, you'll find that, in fact, these polyphenol metabolites went up in the blood of volunteers who ate blueberries. But when the blueberries were mixed with yogurt or with a milk-based smoothie, there was no benefit. There were no uptake of these polyphenol ingredients. So just because it says low fat or low calorie and it's got fruit and probiotics doesn't mean that you're actually going to get the benefit that you think you're getting. More amazing episodes just like this one. Watch now. Studies have shown that adding an avocado a day to human volunteers improve weight loss as opposed to not having that avocado. Just remember, eating fat does not make you fat.