Jorge Pullin: Okay, sorry speaker to this much your subscale who will speak about Korean at the center space times that admit non singular horizons.
Warsaw LQG: Okay. Hello and welcome.
Warsaw LQG: So this is actually a joint work of mine with Priscilla on those key that we're doing here and the faculty of physics at the University for so and
Warsaw LQG: What I want to talk today about at the first the Colonel's and the disk space times they are not the most commonly and contact space dance one considers in GR so to me.
Warsaw LQG: Until I break down some of their properties.
Warsaw LQG: Then
Warsaw LQG: I will introduce the conditions.
Warsaw LQG: Conditions for the removal of the singularity of the clinical singularity of the horizons in the cabinet and the space times
Warsaw LQG: Some discussion about what are those singularities.
Warsaw LQG: And
Warsaw LQG: What is quite surprising result in the same conditions for the removing of singularity and the singularity, the horizon can be extended to the neighborhoods.
Warsaw LQG: Of the non singular horizon. So that will be the next point I will talk about. And finally,
Warsaw LQG: Our work is connected
Warsaw LQG: To another topic. Our team is working on so connected to the type the horizons on the non trivial fiber abundance of our sphere, and I will show some connection of our long singular horizon to this work.
Warsaw LQG: As well as some generalization of our results extending to the accelerator cannot see the space times
Warsaw LQG: In this case, even, can I think the Secret Space Time accelerated, of course.
Warsaw LQG: So what are the seeds are planted the seed stage names.
Warsaw LQG: There are solutions to the vacuum Einstein equations with cosmological constant. And it's a four dimensional family parameter is by
Warsaw LQG: For real numbers, usually, usually interpreted as the muscle my call as the curl parameter connected to the angular momentum of the black hole.
Warsaw LQG: Just like home is on the cosmological constant background.
Warsaw LQG: For the sign is
Warsaw LQG: A sprig can be positive, negative, and there's this one quite exotic parameter called and they're not parameter first discovered in the top not
Warsaw LQG: Space, which was the extension of fortunes swatches.
Warsaw LQG: Black Hole.
Warsaw LQG: And introduction and is not parameter means for us.
Warsaw LQG: Some, some new to political
Warsaw LQG: Considerations has to be taken into account.
Warsaw LQG: So,
Warsaw LQG: Before I begin, I want to talk about some motivation for our work.
Warsaw LQG: So,
Warsaw LQG: The person wanted to investigate just the
Warsaw LQG: The horizons of the current not the Sitter space times
Warsaw LQG: Because you know 13 there was there was some results concerning the solutions to the pit of the equations, which provides us with the geometry of the isolated horizons of time thing.
Warsaw LQG: Any particular one recent ones and paper by the mosquito cleverness garage discusses the family of type the horizons of the topology bundle.
Warsaw LQG: Which
Warsaw LQG: When you, when you pose the vacuum is the equations and provided by industry that I selected horizon solutions.
Warsaw LQG: And those solutions are actually privatized by four parameters so
Warsaw LQG: We conjecture that probably those horizons should be some relation to the current not until they see their space times because the as well and four dimensional family.
Warsaw LQG: But as we discovered later this, the decision was
Warsaw LQG: Quite less obvious than we expected.
Warsaw LQG: So I'm expecting some results only on the sub family of Colonel
Warsaw LQG: colonel colonel this space, things are on the family.
Warsaw LQG: Of the four dimensional family of time the horizons.
Warsaw LQG: So this is the general form of the
Warsaw LQG: Current not under the sitter metric and
Warsaw LQG: There are some
Warsaw LQG: Some things, one should maybe understand just by looking at this metric. So first thing that
Warsaw LQG: It can be seen. That's just a generalization of the previously known like how solutions. So for the for the not parameter equals zero, we will cover the
Warsaw LQG: I think this either spacetime.
Warsaw LQG: And for
Warsaw LQG: A equal to zero and lambda equal to zero. So we actually covered the original thousand ounce metric
Warsaw LQG: So there are a couple of
Warsaw LQG: Things that are quite suspicious. Like, for example,
Warsaw LQG: There are places in the world where we divide in that component, the component of metric so
Warsaw LQG: And my assumption the function P which appears in the beta coefficient has to be has to be positive. It also guarantees us that the
Warsaw LQG: Them signature is Lawrence young so we use, minus, plus, plus, plus.
Warsaw LQG: On the other hand, we also divided by the function kill and dysfunction, kill, as we said later so is corresponding to the
Warsaw LQG: To the kidney crisis on the cabinet.
Warsaw LQG: Space times
Warsaw LQG: Yes. And what's also important about this metric that
Warsaw LQG: As all of the Type D solutions we have algebra have any factors spammed by to committing deals. So we have
Warsaw LQG: The opportunity which is time translation symmetry.
Warsaw LQG: And also the over the five, which means for us that this metric is as a rotational symmetry.
Warsaw LQG: Yes, so
Warsaw LQG: They want to discuss the clinical singularity.
Warsaw LQG: Which stems from the fact that when we, when you consider this one form a defy which appeared
Warsaw LQG: Appeared in the time like metric component
Warsaw LQG: They discovered that
Warsaw LQG: If the L is not zero. So when we have some not parameter
Warsaw LQG: This one phone does not doesn't punish at the pole data equals the pie. So this means for us that the G is not continuous, and the whole semi access Sita equal to buy
Warsaw LQG: And obviously this is a coordinate and dependent statement. So one could choose another coordinate and accordingly transform this singular access. So for example, if you introduce
Warsaw LQG: The prime which is T minus four L fi then
Warsaw LQG: Our one phone transfers and
Warsaw LQG: Such that
Warsaw LQG: It then becomes a prime the data and then
Warsaw LQG: And discussing squared is for us that
Warsaw LQG: This one doesn't management and the semi access data at 1.0 and the genius not continues and
Warsaw LQG: At the at the other similar axes of course for any other choice of the of the team coordinators, for example, and TVs equals t minus two, five should be to find
Warsaw LQG: This one from doesn't punch that the balls balls and the jury is not continuous. In the whole access so
Warsaw LQG: One of our refers to this space times as having the gravitational magnetic monopole while
Warsaw LQG: When this singular axes corresponds to the
Warsaw LQG: To the Mr string connecting the magnetic the bone and as was in that case, we could
Warsaw LQG: Talk about the gravitational magnetic monopole one could choose engage such thoughts, want to switch which access is singular.
Warsaw LQG: And
Warsaw LQG: For the generic values of parameters. So of the mass exposure constant current not parameters the space time you singular in at least one poll
Warsaw LQG: And usually, usually above us.
Warsaw LQG: So there was some propositions in how to remediate those problems.
Warsaw LQG: For example, for example.
Warsaw LQG: Mr considered
Warsaw LQG: Originally I thought not space. So a limit of without care parameter AND CHRIS MITCHELL constant
Warsaw LQG: When we could take
Warsaw LQG: Don't know senior solutions. So one with the singular axis at the southern hemisphere and one and the
Warsaw LQG: Senior axis and the Northern Hemisphere and then we take from the, from the left one. The know them not singular hemisphere and glue it to the
Warsaw LQG: Southern
Warsaw LQG: Of the right
Warsaw LQG: Space and
Warsaw LQG: In this way we arrived at the
Warsaw LQG: At the top, not space time which is which is not senior
Warsaw LQG: At the slides, the slides cost of having a periodic to coordinate
Warsaw LQG: And then the resulting topology is a free spirit times, times around this topic, not space. So our results.
Warsaw LQG: And remained results states that this is not enough.
Warsaw LQG: In the general case in the generic case of digital space times so
Warsaw LQG: One can one can make is doing in the arbitrary case. But then one discovers that
Warsaw LQG: We don't have this
Warsaw LQG: Resulting topology of US frame and our glowing is such that the horizon that the holes are still not differentiable
Warsaw LQG: So what we discovered
Warsaw LQG: Is that the glowing is differentiable
Warsaw LQG: And in fact we by investigation explicitly check that at least twice differentiable, only if this is condition is satisfied. So if we overlap that was to a squared r squared plus two r zero squared.
Warsaw LQG: Where r zero is a radius of
Warsaw LQG: Have a
Warsaw LQG: Clinic Christ on which we want to make non singular
Warsaw LQG: So a couple, a couple of words about the current the sitter horizons.
Warsaw LQG: So, as I told you before the horizons are
Warsaw LQG: Correspond to the to the zeros of this Q function.
Warsaw LQG: Actually just a polynomial polynomial in in the coordinates are so it's for for the recording recording phenomenal so
Warsaw LQG: We have up to four killing horizons and one can check that.
Warsaw LQG: In fact surfaces of our he was to our CEO and our surface surfaces and they are crayons and their clinic horizons never buy this Kinney vector side which we use extensively in this
Warsaw LQG: In our research paper.
Warsaw LQG: Which is just a just a certain linear combination of the two factors that we
Warsaw LQG: That we have. And maybe one important thing to notice.
Warsaw LQG: Is that
Warsaw LQG: This
Warsaw LQG: Killing vector field.
Warsaw LQG: Explicitly depends on the right is on the horizon.
Yes.
Warsaw LQG: So,
Warsaw LQG: Actually, the, the original metric and the original form the metric that I showed
Warsaw LQG: Was singular at the at the horizon. So, one cannot use it to describe the
Warsaw LQG: The Christ and geometry. But what we can do is we can introduce the slightly change call frame. So this coordinate be is an analog over
Warsaw LQG: Of an advanced time
Warsaw LQG: And actually, it just reduces to the advanced time for
Warsaw LQG: For the suitcase.
Warsaw LQG: And also to introduce some some twisted and twisted ankle coordinate
Warsaw LQG: Which is not, not necessarily in the into taking cases we have here and a
Warsaw LQG: And then
Warsaw LQG: You can you can calculate the there's nothing metric that is announcing that are at the horizon. So as you can see
Warsaw LQG: This this troublesome.
Warsaw LQG: Factor of one overkill, which was connected to them the r squared coordinate is no longer appears so our, our metric is is totally fine. And the horizon.
Warsaw LQG: And here we explicitly say that the surfaces.
Warsaw LQG: Of
Warsaw LQG: Constant. Constant. Constant radius.
Warsaw LQG: R zero
Warsaw LQG: Gives us the agenda metric
Warsaw LQG: On the horizon.
Warsaw LQG: And those sigma zero functions and roles, your functions are just the previous functions introduced
Warsaw LQG: With the metric components.
Warsaw LQG: But for the horizon.
Warsaw LQG: Ranges.
Warsaw LQG: So we want to want to introduce
Warsaw LQG: You coordinates on this singular horizon.
Warsaw LQG: And
Warsaw LQG: In fact, this coordinates expo and x ray are the coordinates on the space of now generators of the horizon.
Warsaw LQG: And
Warsaw LQG: When completed with this coordinate style those coordinates.
Warsaw LQG: We just need coordinate systems on the on the horizon and
Warsaw LQG: Great vibe from this curtness
Warsaw LQG: That they are somewhat perpendicular to the
Warsaw LQG: Today, killing better
Warsaw LQG: So, so the size of X to an extra
Warsaw LQG: Damage
Warsaw LQG: So make a particular choice.
Warsaw LQG: Of the talk. Responding to
Warsaw LQG: To be and
Warsaw LQG: X to tune the data and expect to some some combination of the and
Warsaw LQG: And fight to the connected with
Warsaw LQG: Collectors also to this Amiga which define the
Warsaw LQG: The killing machine vector
Warsaw LQG: And then we have this
Warsaw LQG: This new
Warsaw LQG: Horizon metric in this new ordinance when what it is even. It's even easier to see that he said that Jeanette degenerate metric on them on the horizon.
Warsaw LQG: And one thing I should also point out is that
Warsaw LQG: Will probably use this fact that
Warsaw LQG: In this in this coordinate the tower is just side.
Warsaw LQG: So it will simplify some some notation.
Warsaw LQG: Yes, and
Warsaw LQG: Maybe some intuitions about once or this clinical singularity.
Warsaw LQG: And responsible, so we can we can consider section of their horizon and in many cases, one one expect this section of the horizon to have the topology of the spear.
Warsaw LQG: But there might be some some problems in the Pops.
Warsaw LQG: In fact, our, our previous metric
Warsaw LQG: Was analytic everywhere.
Warsaw LQG: Except for me from the, from the post. So, so what I want to survey to study this metric. The only suspicious points are
FAU Erlangen: Me. Can I make a comment is, so actually this figure, which is the next slide. It is the it's not a slice of the horizon. It is the projected space. This is the space of the knowledge generators and this is exactly this metric from the previous slide.
Warsaw LQG: Yes, yes.
Warsaw LQG: So what can go wrong in the polls, is that we, we can have some kind of some kind of wedge this this web this
Warsaw LQG: Corresponds to this clinical singularity. And one way of saying it is
Warsaw LQG: Is that we simply chose chose the range of the angular Coordinates. Coordinates wrong.
Warsaw LQG: And
Warsaw LQG: I would like their methods is based on considering and such loops.
Warsaw LQG: Around
Warsaw LQG: Circles circles around the symmetry access and in the case of the
Warsaw LQG: Way from far from the pole, this, this ratio should obviously be to buy, but there might be some problems in the post. And then we'd like to impose the conditions.
Warsaw LQG: That duration of circumference to the radius in the post will also be
Warsaw LQG: Like on the to secure. So to buy
Warsaw LQG: Yeah, so that's exactly. That's exactly what I'm going to talk now. So, so far we haven't made many assumptions about the the nature of the coordinates x to an extreme. And in fact,
Warsaw LQG: In fact, we didn't assume that there are spiritual coordinates.
Warsaw LQG: So explain
Warsaw LQG: Expands
Warsaw LQG: expands on some interval close at the one side, which is two pi time. See, and later on finding this the suitable see constant will allow us to remove that
Warsaw LQG: The single the
Warsaw LQG: Clinical singularity.
Warsaw LQG: So,
Warsaw LQG: So actually made a little bit of assumptions that, but there may be not assumptions, because they can be seen from the, from the form of the metric that x to y equals zero corresponds to the single bond and
Warsaw LQG: So our poll is a single point, and I still equals to some other concerns our circles on this.
Warsaw LQG: Space of no generators
Warsaw LQG: Now we have to define
Warsaw LQG: Went to consider the circles of a constant stone and define for them.
Warsaw LQG: There's a conference which is an integral of the
Warsaw LQG: Of the of the components.
Warsaw LQG: And thank you, free free
Warsaw LQG: Along the circle and then we want to introduce the radius which we calculate in such a in different ways in different calls different polls and this choice is made so that
Warsaw LQG: So that the radius calculated along this
Warsaw LQG: And this
Warsaw LQG: Money for that. You want some beautiful
Warsaw LQG: Spirit is calculated along along just disappear and
Warsaw LQG: The radius on the northern
Warsaw LQG: Hemisphere.
Warsaw LQG: should train to zero as we approach the pole and the same goes for the rate is defined the southern hemisphere. So it should also shrink to zero when we approach approach the boss.
Warsaw LQG: So as I mentioned earlier, our condition.
Warsaw LQG: Our condition that the radius that the ratio of the circumference to the radius.
Warsaw LQG: Has to be checked in, in the suspicious points of our metrics on the pose and they simply corresponds to taking the limits of the circumference and their ideas.
Warsaw LQG: In boats boats and
Warsaw LQG: Making strength of this limit has to be to buy so that we will cover the result from a sphere.
Warsaw LQG: And so because in the in the integration.
Warsaw LQG: Interval appears. This is Kelly constancy. And he says, he says the the condition on this is getting constant. So for the. So the first thing is that we want.
Warsaw LQG: That we want this rescaling constant to be to be the same in both balls, so that's that's what this first equations equations stems from and then we see that there's Kelly constantly simply one over the, over zero so that we have to fight for example in in this in this Poland.
Warsaw LQG: If this part satisfies
Warsaw LQG: The first constraint and we also had
Warsaw LQG: To to find and the brief reminder about dysfunction be so in general this condition this condition.
Warsaw LQG: Should depends on the on all of the parameters of the Colonel's and the Sitter space time. So we have
Warsaw LQG: Except from like from them. So
Warsaw LQG: So we have great news. Not parameter
Warsaw LQG: Awesome. Sure appears because mariska constant
Warsaw LQG: So in five our, our constraint is quite with the study because we will cover the previously known.
Warsaw LQG: This no solutions which are which do admit Monsignor our horizons. So first, if one says, because much constant to zero then then be PLC one piece of pie are equal. And then the only choice to make this distraction. One is to
Warsaw LQG: Either have l equal to zero and so that the squad takes care of the miners or have a equal to zero. And those corresponds to either current or 12 not
Warsaw LQG: Space times
Warsaw LQG: Similarly,
Warsaw LQG: Similarly, if one
Warsaw LQG: Takes equal to zero.
Warsaw LQG: So some curve and visitors FaceTime or
Warsaw LQG: Equal to zero and some some London so told not to disturb space time. Then we also have
Warsaw LQG: Feels very well be of bison, so it's good that our, our constraint recovers the non non singular solutions.
Warsaw LQG: But there's one more choice. One more solution to this equation.
Warsaw LQG: And this distortion requires a couple of assumptions. So it requires that all of the
Warsaw LQG: All of the parameters are non zero so
Warsaw LQG: So, so, in particular, we cannot have any symptoms are
Warsaw LQG: Not solution without customers are constant
Warsaw LQG: And this is this is the constraint. I was, I was thinking earlier and it might be also
Warsaw LQG: Intuitive to see why why it doesn't admit zero because Mitchell, a constant because we we actually, when do I mean the rescaling constant, we, we did. Bye bye. Bye.
Warsaw LQG: Then our, our constraints has are zero and we like to probably would rather
Warsaw LQG: Maybe not in general the the space time is prioritized by Master
Warsaw LQG: actually solving solving
Warsaw LQG: We can solve and the queue of our zero equals zero and get and get the, the mass of the black hole.
Warsaw LQG: So a couple of remarks about the geometry of the space of the now geneticists decry some
Warsaw LQG: Some now.
Warsaw LQG: Now the coordinates.
Warsaw LQG: X two X extreme
Warsaw LQG: Has this range, when we determined determined the rescaling constant is one over P of zero and one could really nicely.
Warsaw LQG: describe those those horizons, because we can explicitly calculate the value of the other area so
Warsaw LQG: It's somewhat connected to the reason why we cannot have a negative customer Chad constant
Warsaw LQG: Because as long can see then the area would be negative. And in fact, the this condition for lambda
Warsaw LQG: Complaints only, only some positive numbers and then the squares of the parameters. So, so we have no no singular bicker not emptying the Sitter space times with
Warsaw LQG: With customers with negative because much a constant.
Warsaw LQG: So,
Warsaw LQG: We explicitly check that. Our philosophy is at least twice differentiable at the polls. So, so that the metric that the components of the metric or at least twice in different ways differentiable
Warsaw LQG: But contrary to the for example current horizons.
Warsaw LQG: They do not have
Warsaw LQG: The topology over to spear times, times our. So actually, our horizons have some of the apology have some non trivial.
Warsaw LQG: Fiber been over a stone, such as such as, for example, hope bundle.
Warsaw LQG: And there is also some
Warsaw LQG: Some tricky results, some tricky.
Warsaw LQG: parts of ourselves, is that
Warsaw LQG: This condition explicitly depends on the on the radius of the horizon.
Warsaw LQG: So,
Warsaw LQG: In general, if we if we have one
Warsaw LQG: Singular horizon.
Warsaw LQG: Then we could ask, what about the up to free other horizons. And in fact, this, this, this condition tells us
Warsaw LQG: That if you make one of the crisis one singular, then there's no no possibility of making other horizons non singular
Warsaw LQG: Yes, so the next. The next thing we consider was where the the neighborhood of the horizons and
Warsaw LQG: To do that we introduced the space of the orbits the orbits of the Kinect oxide and directly the horizon.
Warsaw LQG: And to do that we chose
Warsaw LQG: Chose the coordinates, such as exciting equals to some detail. And in fact, the
Warsaw LQG: If the code is that I'll show you in a minute quite similar to those for the horizon.
Warsaw LQG: And then if we if you can write the metric in this
Warsaw LQG: New Horizons in this form.
Warsaw LQG: Then
Warsaw LQG: Then there are a couple of nice nice things about about this way of writing the metric. So for example, this Q part so que que ha the X the X the X j correspond to the metric on the space of the orbits of the better side. So one could maybe better understand what this metric is
Warsaw LQG: Is if we consider that the the space time
Warsaw LQG: Is
Warsaw LQG: Created by the by the flows on the kinetic side and
Warsaw LQG: Each
Warsaw LQG: And this low corresponds to
Warsaw LQG: The clinic observers and then
Warsaw LQG: Then the skill measure space time distance between between some neighboring kingdoms observers.
Warsaw LQG: Along alongside
Warsaw LQG: So some other some other part is
Warsaw LQG: This one from ditto plus and the dog is
Warsaw LQG: Divided by Jane Doe.
Warsaw LQG: The x
Warsaw LQG: Which is
Warsaw LQG: Which we call the rotation connection one form.
Warsaw LQG: Because it is somehow connected to the to the rotation of the space and also this one firms actually a connection on the
Warsaw LQG: Connection form on the hub bundle which
Warsaw LQG: Of which the our horizon structure.
Warsaw LQG: So,
Warsaw LQG: What's the benefit of writing that the metric in this way. So all of these
Warsaw LQG: All of these objects that they described as well that loves function they have they have
Warsaw LQG: Geometric geometric meaning besides some some coordinated expression. So it's, it's actually good because our considerations about the horizon. We're also
Warsaw LQG: Kind of geometry, because the horizon is a it's an object which is defined geometry geometry and not not in some particular coordinates.
Warsaw LQG: And so this is this is the analytical approach.
Warsaw LQG: In the neighborhood of that the crisis.
Warsaw LQG: Some will we make some particular choice of the of the coordinates.
Warsaw LQG: And this choice is quite similar to the
Warsaw LQG: To the choice for the horizon. So, and before we had Tao equal to some
Warsaw LQG: Advanced code and advice advanced and coordinate. Now we have simply simply team and also there is x one, which is just the radius.
Warsaw LQG: And the rest is is the same as in the horizon case.
Warsaw LQG: And then we can ride the, the original American in this new form.
Warsaw LQG: For me, which we
Warsaw LQG: Extract
Warsaw LQG: The form of the metric on the space of the orbits.
Warsaw LQG: So if you want this this metric to measure distance. Distance between the kink observers, we have to make sure that that we are somehow using the frame which is tangent to the
Warsaw LQG: To the, to the side. So we are introducing introducing new frame this hearted frame.
Warsaw LQG: That exactly satisfies satisfies our property.
Warsaw LQG: Property that is it's it's perpendicular to
Warsaw LQG: The side and
Warsaw LQG: When I say live and this this metric. Q.
Warsaw LQG: On the space of the of the orbits.
Warsaw LQG: And obviously, it has some properties that I would like to discuss now. So first of first of which is the
Warsaw LQG: The signature.
Warsaw LQG: So these components Q human lung.
Warsaw LQG: Close to the horizon, just the non extra mile for aizen which might be some which actually makes a difference here. What kind of Horizon, we do consider
Warsaw LQG: Which wasn't the case, the horizon case on the the previous metric was in qualifying for the for the external horizons or so when when the R zero was a multiple of the function Q
Warsaw LQG: Now we only consider the unknown extreme cases.
Warsaw LQG: So close to the horizon, the cumin. One is proportional to minus g psych side. So when the time is is time like
Warsaw LQG: This component is positive and the other way around. When is space like
Warsaw LQG: To to to buy the assumption that the P is positive. You see, it's also positive everywhere.
Warsaw LQG: And the component
Warsaw LQG: To
Warsaw LQG: The component, feel free. Three also requires us that we are
Warsaw LQG: Supposedly, but only one we are close to the non extra mile horizon.
Warsaw LQG: So this can probably be better seen from from this
Warsaw LQG: Explicit
Warsaw LQG: Form well
Warsaw LQG: Then what we what we did here was just divide by my dysfunction kill and if
Warsaw LQG: In the denominator, the negative part
Warsaw LQG: If it vanishes. Then, then this be science squared 00 squared sigma divided by sigma squared, it's positive. So then we would have
Warsaw LQG: We would have
Warsaw LQG: You feel free positive
Warsaw LQG: And when we consider this this denominator
Warsaw LQG: It goes to zero when we approach the horizon, only if the
Warsaw LQG: If the
Warsaw LQG: R zero
Warsaw LQG: Corresponds to the non singular person.
Warsaw LQG: Otherwise this this second second part. And denominator approaches some concept on even can
Warsaw LQG: Even can
Warsaw LQG: Blow up and this is this is a result
Warsaw LQG: This stems from the fact that the
Warsaw LQG: The metric and and the the space of the
Warsaw LQG: Of the purpose of the killing horizon as well as you find on the way.
Warsaw LQG: If you are far from the horizon.
Warsaw LQG: And this is also some some it's mirrored in the fact that the denominator of the component queue for free. It's actually proportional proportional to the to the length of the
Warsaw LQG: Of the vector side.
Warsaw LQG: So,
Warsaw LQG: If we are close to the to the extra mile for eyes on this component
Warsaw LQG: Feel free. Free to approach the
Warsaw LQG: The component cube for free from the metric on the on the horizon and
Warsaw LQG: That would be fine for us. We have just some extension of the geometry from the horizon to the neighborhood, but
Warsaw LQG: There, there might be some services which are not killing horizons.
Warsaw LQG: And
Warsaw LQG: All the way to the the norm on tech side also vanishes. So that's kind of a natural range of the applicability of our of our metals.
Warsaw LQG: Yes, so. So, the important part here is that that this metric tends to the present metric. If the R zero is a single
Warsaw LQG: Singular it corresponding to the non western horizon.
Warsaw LQG: So the considerations from the previous slide, it has to the conclusion that
Warsaw LQG: First, we, we should consider here on the, the non external four zones.
Warsaw LQG: So are in order to to assure that our signature is either plus, plus, plus or minus plus plus one. What's really important for us is that, and that
Warsaw LQG: There are two classes at the end here.
Warsaw LQG: And also,
Warsaw LQG: There are a couple of statements that dependent on the fact that we are close to the horizon. So in fact,
Warsaw LQG: Our metric
Warsaw LQG: Or a metric is defined on the on some on some future or past neighborhood on the horizon.
Warsaw LQG: And this
Warsaw LQG: Is a short assures us that
Warsaw LQG: That excite does not doesn't vanish.
Warsaw LQG: And
Warsaw LQG: And this component to free free is well defined.
Warsaw LQG: So,
Warsaw LQG: Again, we want to
Warsaw LQG: We want to consider.
Warsaw LQG: Want to consider whether this metric is is analytic and of course it should be analytic everywhere was defined, apart from the posts on the same case as the horizon.
Warsaw LQG: And then the only the only
Warsaw LQG: The only part of this metric, which can be which can be singled out the post have some problems is the pullback of this metric to to the surfaces.
Warsaw LQG: With the surfaces of X one constant but not equal to the horizon radios will, which was the case that when considered earlier.
Warsaw LQG: For this for this two dimensional matrix.
Warsaw LQG: We can look in again use the framework of considering loops.
Warsaw LQG: loops around
Warsaw LQG: The rotational axis.
Warsaw LQG: So the, the circumference of the circle and the radius are defined just just in the same way as for the horizon.
Warsaw LQG: And what we somewhat surprisingly discovered is that when we apply this framework of of loops and we impose this condition that
Warsaw LQG: And that the ratio of circumference to the radius should be two pi in the post that we recover exactly the same condition is for the horizon, so that P zero equals to some friction and then p of by
Warsaw LQG: And
Warsaw LQG: So what it means is that if we can some some
Warsaw LQG: Parameters that I've made no single our horizon, then at the same time simultaneously those parameters.
Warsaw LQG: Remove the singularity from the, from the future or past neighborhood of the horizon. And what's also important to stress here is that the our rescaling constant is still simply one over
Warsaw LQG: You know, and the function PR zero does not depend on the value on the coordinate or here on the x coordinate, so
Warsaw LQG: So,
Warsaw LQG: So on the all of the surface. That's funny. The, the neighborhood of the horizon. We use exactly the same rescaling as the as the horizon, so we can make
Warsaw LQG: The horizon and the the past or future neighborhood one singular simultaneously with their horizon.
Warsaw LQG: Yes. So, so
Warsaw LQG: So that was our main results that we will cover this
Warsaw LQG: Condition for the non singularity and then salted
Warsaw LQG: So it's so you can see the first, the general case of the kernel space time privatized by massacre perimeter not parameter and lambda
Warsaw LQG: We examine the geometry of the non generic horizon. And what we found is that
Warsaw LQG: Consistent with the previous previous statements in the literature, generally, but the general generic values of the parameters the singularity is
Warsaw LQG: Non removable in at least one of the and the polls.
Warsaw LQG: Then we will post this condition down to the radius that the ratio of the circumference
Warsaw LQG: And the ratio of the loops around the rotational axis as to tend to two pi.
Warsaw LQG: In the polls. These gave us the condition for removing the singularity, which we solve and obtain the new and new solution. So this constraint lambda equals two free over a squared plus two squared plus two, or zero squared.
Warsaw LQG: So that was our new result and also our conditional recovered the previous know
Warsaw LQG: The previously known singer.
Warsaw LQG: Solutions. So
Warsaw LQG: Curve the theater space times and not the center around like to sit there your space times
Warsaw LQG: And
Warsaw LQG: Just think, fear is that there is still this
Warsaw LQG: somewhat surprising for us part that
Warsaw LQG: We can also we can make at least at most one of the horizon, no singular at the time because our constraints explicitly
Warsaw LQG: Depends on the value or the radius of their horizon.
Warsaw LQG: So next we we consider also
Warsaw LQG: The neighborhood of the non singular prisons. So employed the framework of the
Warsaw LQG: Space of the
Warsaw LQG: orbits of the healing.
Warsaw LQG: Vector developing the horizon.
Warsaw LQG: And we found that it has a good properties in the known external case.
Warsaw LQG: But we have to restrict ourselves to some small neighborhood of the horizon.
Warsaw LQG: It should be zero here.
Warsaw LQG: And then our metric metric
Warsaw LQG: On the model space is well defined and has the signature. Not that we want actually
Warsaw LQG: So our neighbors was his the topology of a sweet time some some interval around
Warsaw LQG: Around the horizon. And in fact, each of the horizon has the topology of the abandoned.
Warsaw LQG: Over, over a stone.
Warsaw LQG: So,
Warsaw LQG: I will research as I, as I said in the beginning, originated from from the quest to find
Warsaw LQG: To to embed
Warsaw LQG: All of the type deep, but from the path of typing and vacuum isolated horizons, which were
Warsaw LQG: Which were
Warsaw LQG: Found in some more abstract way we wanted to
Warsaw LQG: To know how they correspond to into the unknown metrics. How are they embeddable
Warsaw LQG: And Don'ts horizons on and non trivial to political bundle.
Warsaw LQG: There were four dimensional family.
Warsaw LQG: And because
Warsaw LQG: Our
Warsaw LQG: Our horizons are not singular only if we pull someone constraint on the cosmetic constant. We only have a three dimensional sub family. So the next quite obvious question is, was about the remaining time horizons and
Warsaw LQG: We what we consider next where the accelerated cannot enter the Sitter space time
Warsaw LQG: So one course, one can interpret the space time as a min similarity to the case of the current not under this space time but the black hole is also some some acceleration
Warsaw LQG: Along the singular access and does it gives us a new degree of freedom.
Warsaw LQG: Which we conjecture that can be
Warsaw LQG: Connected to the fourth fourth dimensional and fourth dimension on this family have better update Dean biking horizon some some mantra that bundle.
Warsaw LQG: And in fact, when we consider this case we arrived at exactly the same condition.
Warsaw LQG: For the removing removing of
Warsaw LQG: Of the singularity.
Warsaw LQG: We employed the same framework we arrived at the end result that was formerly the same
Warsaw LQG: So remember this condition constrained with with p zero P of by. So, this condition is the same.
Warsaw LQG: But
Warsaw LQG: The form of this P function diverse it's somewhat more complicated for the accelerated case.
Warsaw LQG: And we also solve this
Warsaw LQG: So this constraint and right there's some four dimensional family of accelerated current not on data center space time that meetings non singular horizons and neighborhoods and
Warsaw LQG: Here there's an additional benefit that the constraint that we should have a positive because Minister constant, this
Warsaw LQG: Is relaxed. So we don't mean some some negative customer for constant also
Warsaw LQG: So,
Warsaw LQG: More details about our methods about our resolve.
Warsaw LQG: should appear in the upcoming paper some some time, we have this mouth and
Warsaw LQG: It was not the answer questions. Thanks. Thank you for your attention.
Penn State: No. Can you hear me.
Warsaw LQG: I can hear you now.
Penn State: Okay, so this is like a couple questions. The first is
Penn State: If you go back to slide 14
Penn State: Yeah, so you're looking for known singular certain non non solutions are, I guess.
Penn State: It's like 30 you're looking at non singular solutions solutions with non single horizons. Right.
Penn State: 13 or 1413 you're looking at general conditions you wrote down the slide. You're all done. And then in the next slide.
Warsaw LQG: 14
Penn State: Years give non non singular solutions.
Penn State: Why so what happens to
Penn State: The Singularity persistent. What's the story.
Warsaw LQG: Yes incarnate. You cannot remove the singularity. Okay.
Thank you.
Penn State: The product question is the following that. I don't know if I understood the main message but i mean you know your result is that
Penn State: Let's consider this
Penn State: This for parameter family of solutions. And of course, it is of interest to consider the case where the generic case where none of the parameters is zero right
Warsaw LQG: Yes.
Penn State: And then in that case, sort of,
Penn State: Your results implied that whether the cosmology of constant is positive or negative.
Penn State: There is no way to get rid of.
Penn State: The conical singularity on all the prizes.
Warsaw LQG: Yes. So our clustering depends on the radio surprising sober, I
Penn State: Might be able to
Penn State: Choose your parameters and I mean if you like can be thought of as r&r or not depends on everything else.
Penn State: So you can choose your parameters MN lambda
And
Penn State: L an A. Yeah, so you can choose those parameters, such that
Penn State: Such that one or eyes and everything is fine, but then something will be wrong adopt another reason normally one sort of
Penn State: Physically one is interested in solutions which don't have Monica singularities. Anyway, that was the main point of it. Is there any other people. Right.
Penn State: Because of our clinical singularity is that means that there is a Einstein's equation or some stress any answer which corresponds to membrane no order.
Penn State: So then
Penn State: My take home message I took it to be that
Penn State: The case with all four parameters non zero is physically not interesting. Am I missing something.
Warsaw LQG: In
Penn State: This case, irrespective of the sign of the cosmos actual cost and
Penn State: It will be Hanukkah singularity is somewhere in the space time
Warsaw LQG: It's so surprising results for us. Also, we also don't fully understand
Warsaw LQG: Why this is the case.
Warsaw LQG: Especially the the reason
Penn State: Why this is the case, I'm just saying that if
Penn State: Assuming that this is the case, which is your result.
Penn State: Then I will just conclude that, that, that, that is, it implies that therefore
The
Penn State: Space times with all for panorama does not equal to zero or physically not interesting.
Penn State: Because they will always have some why single I mean this sheet of singularity right this conical singularity between
Penn State: idolizes
Warsaw LQG: Results in place, right.
Penn State: So you can hear.
Penn State: Me.
FAU Erlangen: And I can I can I help much, much, you can also
FAU Erlangen: Yeah please elaborate on your question. Okay, so, so this is the zero to order conclusion that that this singularity cannot be removed all space. I mean, now this
FAU Erlangen: peculiar property. Is that still for some special a choice of the cosmological constant
FAU Erlangen: In neighborhood. One of the horizons becomes non singular and and its neighborhood is also
FAU Erlangen: So, well, maybe it's not enough for you if you would like all the space time to be
Penn State: Yeah, because that's what anybody would say, right, anybody would say that if I want to take the space time. Can you speak to be interesting. It is not enough that one neighborhood is this
FAU Erlangen: So it depends on who we're talking to, if somebody is not interested in
Nothing interesting
Penn State: Exactly. The question I wanted to answer.
FAU Erlangen: It is in between subtlety that are those horizons in
FAU Erlangen: This Theater, which in some special for the values of cosmological constant are in non singular and also they are surrounded by non singular
FAU Erlangen: Neighborhood. But let me also say something more. So in fact, we show that we're results to people who who are who wrote earlier some papers on not
FAU Erlangen: Space times is actually our statement that generically those horizons have this unremovable clinical singularity is not something which is commonly
FAU Erlangen: Believed and understood, people usually think that the singularity follows because of this non trivial topological character. And if we properly.
FAU Erlangen: Glue space time then we grew one singular part was another non singular part and we
FAU Erlangen: Don't have singularity at all. This is I think Mark Mars, send us the reference to some paper in which
FAU Erlangen: This is explicitly stated, however, nobody proves this is just written as some remark that obviously if we create space I'm that we don't have
FAU Erlangen: This clinical singularity. So, so the generic. So this take home message which you want to take it is also something something new because you may find statements in literature that actually there is no clinical singularity. If you properly. A extend this space time by this political glue
Penn State: So is that true that, in this case, then if you
Penn State: Have appropriate political identifications you can remove
Penn State: The clinical singularity everywhere in space time when in the, in the case that is on the, on the, on the slide right night right now, which is that none of the parameters is equal to zero.
FAU Erlangen: Know, I will resolve shows you can have
Penn State: Yeah, so right so sauce. So I'm still don't understand. You said that you talked to some people and they were saying that this is a space types of physical interest and I'm very curious to know
Penn State: What would be the physical application.
FAU Erlangen: Well, this is a good, good question. I, I'm not sure.
Penn State: So I just wanted to say that, you know, many, many years ago, I looked at also the the see metric, which has clinical singularity generically. It has parameters which is mass acceleration. And you can also enter the charge. And if you're all these see
Penn State: Charges nonzero bed. In fact, you can do exactly what we have here on this slide. Maybe you can choose a particular value of charge, which is given by the other parameters, such that there is no clinical singularity.
Penn State: But in that case it is there's no clinical singularity anywhere is based on that because they just to black hole. So it's not so complicated, like a cosmological horizon some black hole arises and so
Penn State: So I think that's that's the reason why I'm very interested in this because it looked to me that it has some features like the charge symmetric
Penn State: Okay. And the question was what along the way, did you learn anything striking or interesting or just as a side remark about the courtesy to our case.
Penn State: You know those eloquent zero case.
Warsaw LQG: So our patients and this is constantly reminds us that they don't work for the lambda zero and so
Warsaw LQG: So I think that no
Okay, thank you.
Warsaw LQG: Thank you.
Gravity Group, Univ. Wroclaw: Hello. So what is something maybe from the side. Let me get a little car from rosov
Out.
Gravity Group, Univ. Wroclaw: Okay, so I would like to learn more about your constraint between cosmological constant, this, this condition with our zero square where it came from. You said that gluing is not enough. So this is some kind of differentiable result, I don't know, like how to this and where you got this
Warsaw LQG: So you're talking about this condition.
Gravity Group, Univ. Wroclaw: Yes, yes.
Warsaw LQG: Yes, exactly.
Warsaw LQG: So, it
Warsaw LQG: Comes from solving this this equation.
Warsaw LQG: This constraint which which to arrive at by
Warsaw LQG: Requiring that
Warsaw LQG: The limit is to buy
Gravity Group, Univ. Wroclaw: So this is just some kind of consistency condition between on two poles.
Gravity Group, Univ. Wroclaw: Yes, somebody with some kind of light colored say
Gravity Group, Univ. Wroclaw: One expression multiplying P pie and another expression multiplying p zero
Gravity Group, Univ. Wroclaw: So it's some kind of consistency condition between the polls, because let me say a couple reminders, because I was playing in the past about hope not. Space name. He also had one slide with my
Gravity Group, Univ. Wroclaw: My work. So basically, I noticed that behavior of this strings to be somehow like representing a rotation like basically you can have another set of killing vectors and just consider that you have a constant angular velocity and then that explains somehow why this taupe
Gravity Group, Univ. Wroclaw: List is not parameter actually is very similar to the care parameter in the formulas. Somehow it like a represents that projection on busy access even
Gravity Group, Univ. Wroclaw: When you look closer to the formulas and basically along the way I wanted to, to, to leave this to canonical a clinical singularities basically
Gravity Group, Univ. Wroclaw: Represent the black hole peers five at cosmic string, which was rotating or to cosmic strings rotating
Gravity Group, Univ. Wroclaw: And then basically, I wanted to leave this space them as it is hoping that I will get some kind of maybe another kind of condition consistency condition which would like avoid glowing about maybe along the way, having something. So you have some kind of condition which looks
Gravity Group, Univ. Wroclaw: In the, in the service philosophy, but you want always to remove this severity. So it's like Mark
Warsaw LQG: Is also, we do want to make this glowing
Warsaw LQG: Hmm, contrary to
Warsaw LQG: What you're trying to do.
Gravity Group, Univ. Wroclaw: I just wanted to do, like, see what what is possible to learn more about this metric, as it is right along the way. I don't have any conclusions, but I notice couple of of really strange kind of papers.
Gravity Group, Univ. Wroclaw: Really like the rotation rotating character of this kind of things, not only the kind of like mass parameters stuff.
FAU Erlangen: I would like to add something to the reply which magic, magic sauce. So the question was what is actually non singular if we still can glue do gluing of two to two sections. So, in
FAU Erlangen: Our space time couple logically has structure on this hub bundle times some to other dimensions which I can tractable, so they don't make topology. They don't change anything, and now
FAU Erlangen: There are some objects on on bundle which are engaged set gauge dependent or section dependent. However, there are some structures which are
FAU Erlangen: Independent and would we consider is the space of fibers and the space of fibers is geometric object that this is a choice of slice independent
FAU Erlangen: And if this space of fibers has a problem has some differential ability then choosing sections is not helping so so space of fibers is this invariant object.
FAU Erlangen: Is defined in independently on on choice of coordinates. So we consider. So the bottom line is we consider some property, which is independent of choice of coordinates.
Jorge Pullin: Any other questions.
FAU Erlangen: Actually I have I have still
FAU Erlangen: Answer, maybe, maybe some extension of my answer, which I gave to buy. So I buys
FAU Erlangen: A conclusion from this, from this result was that it because, as we say space time can be made non singular only in the
FAU Erlangen: At most, for one horizon and only neighborhood of this horizon. So it is not physically. It cannot be physically interesting I think this is not something which we check that.
FAU Erlangen: Actually, it can happen for the outer most horizon. So if we if this outermost horizon is non singular, then it's possible that maybe we can extend this
FAU Erlangen: We can remove singularity, all the way from from this horizon to to to all the for all the values of
FAU Erlangen: Of our, of course, still those cabinet space times have they another simplistically flights. So, so I can say, we can do it all the way to describe, but we can do it all the way to infinity of our perhaps this is a construction.
Penn State: Thank you. First of all, thank you very much for your clarification, with the previous clarification, which is that that what you're doing is on the on the space of
Penn State: Generators. And so that really is very important time in that. And that's why
Penn State: Not because if the space generators singularity is much more serious than some cross section is it so thank you for your last answer. The point is that such a space, time, and even
Penn State: The sitter.
Penn State: Would have many several prizes. So just looking so what will happen is that just to get out to most wonders interesting that, then the exterior of the outermost is perfectly fine, but then there will be
Penn State: Kind of kind of comical singularity is in between the two other horizons. Right. I mean, you can you go through your reasons. And so the statement is that they'll be conical singularity in the two other lines, but maybe what you would like to say something like
Penn State: For some reason, physical reason you might be able to argue that
Penn State: Only the
Penn State: The part outside the outermost horizon is what is of interest, but you have to check if that horizon as a cosmological horizon, or is it a black hole of horizon, or what kind of horizon, it is or is a new kind of. So thank you very much.
FAU Erlangen: Yes, yes, that's true, that this is outermost horizon may well be a cosmological horizon and this is also something we don't know for sure.
Jorge Pullin: Any more questions.
Gravity Group, Univ. Wroclaw: Whether you want to have you look at hyperbolic version of because
Gravity Group, Univ. Wroclaw: The paper by Robert month where they consider playing with hyperbolic coconut and some kind of special limits were not parameter has like a cosmological constant
Gravity Group, Univ. Wroclaw: Expression and then there is some kind of duality with like a care. So this hyper bowling one have minuses in some places and that allowing you to kind of certain limits that something is going away. But basically, there is a relation between not parameter and cosmological constant
Gravity Group, Univ. Wroclaw: So that's it. Have you, have you look at it, summer.
Gravity Group, Univ. Wroclaw: Consider if it doesn't change something, if you have another relation between an a parameter and cosmological constant
Warsaw LQG: Well, we haven't looked at this work actually so so
Warsaw LQG: On the fire on the answer.
Warsaw LQG: Would change.
Warsaw LQG: If you can send us that will look at it. Okay.
Warsaw LQG: This paper.
FAU Erlangen: And didn't work in that work is the shape of slice of the horizon is all the while this horizon is a bundle over to or
Gravity Group, Univ. Wroclaw: Over some some
Gravity Group, Univ. Wroclaw: I just like, I don't really need it, but it was not care, too. I like Ultra spinning terror on this another side of duality. So I don't know exactly what is whether you send the paper and hopefully
FAU Erlangen: Thanks.
Jorge Pullin: Anything else
Jorge Pullin: Okay, let's thank the speaker again.