In 8:01: Let p, q be a two points. Then the set of the causal path from p to q with the finite Euclidean length is compact. In the proof of this statement 8:01, to me, it can be also applied to the example of non-compact path, x(t) =sin(n pi t), because sin(n pi t) is also included in the square. But he said such a famility is not compact. Where do I missunderstand? I know that if we use the procedure of 8:01 for x(t) =sin(n pi t), then the limit of the resulting subsequence dose not continuous. So, to show that the limit is continuous, we need to further argument,,,maybe.
sin(n pi t) is included in the square, but it's not causal for sufficiently large n (slope cannot be greater than pi/4 from vertical at any point, otherwise the particle will exceed the velocity of light). In the Euclidean picture, causality is violated because sin(n pi t) path length will be > sqrt(2) for sufficiently large n.
And so kids that’s how the universe got started. It’s pretty obvious isn’t it? Thanks for listening. I must get in my flying saucer and go home now. Peace.
Is there a possible topology to say a family of causal paths connecting between two fixed ponts p,q is compact? To ensure that a limit causal path is twice -differentiable for the geodesic equation, I guess the topology shoule be the sup norm up to second oreder derivative plus the holder norm of second derivative.
I wonder what goes wrong in the higher dimensional scalar-tensor solutions of string theory which are non-singular, but contain p-branes, like the 3+1 dimensional membrane black hole of type IIB supergravity.
you mean like 10:49? A cone would not work because “p” is not affected by the corners of a cone because they are spacelike to “p”. Remove the corners from your cone and you get a diamond
@@qz1771 It seems in 3:15 he mentions future cone so there are two cones stacked onto each other, the future one being reversed (this I learned from Penrose video). So whatever comes to p must come via a cone and whatever leaves q must leave as a cone. The surface is light propagation and inside are speeds less than light speed. Since the surface is never-ending, they will intersect. But why is 2D diamond depicted and not 3D cones?
@@Burevestnik9M730 Hi Jovan, both a 2D diamond and 3D cones (arranged so that the top one is inverted) would work. A presentation can only display 2D images, however. Moreover, your description of 3D isn't completely faithful still since we are describing 4D processes. After some getting used to the 2D representation will get familiar!
Keith butler are you referring to them covering the subject many times or some personal case of seeing the same thing many times or is there a personal vein to this.
In 8:01:
Let p, q be a two points. Then the set of the causal path from p to q with the finite Euclidean length is compact. In the proof of this statement 8:01, to me, it can be also applied to the example of non-compact path, x(t) =sin(n pi t), because sin(n pi t) is also included in the square. But he said such a famility is not compact. Where do I missunderstand?
I know that if we use the procedure of 8:01 for x(t) =sin(n pi t), then the limit of the resulting subsequence dose not continuous. So, to show that the limit is continuous, we need to further argument,,,maybe.
sin(n pi t) is included in the square, but it's not causal for sufficiently large n (slope cannot be greater than pi/4 from vertical at any point, otherwise the particle will exceed the velocity of light). In the Euclidean picture, causality is violated because sin(n pi t) path length will be > sqrt(2) for sufficiently large n.
These are basically the ideas for which Penrose got his Nobel
And so kids that’s how the universe got started. It’s pretty obvious isn’t it? Thanks for listening. I must get in my flying saucer and go home now. Peace.
Is there a possible topology to say a family of causal paths connecting between two fixed ponts p,q is compact?
To ensure that a limit causal path is twice -differentiable for the geodesic equation, I guess the topology shoule be the sup norm up to second oreder derivative plus the holder norm of second derivative.
Ed at his best
I wonder what goes wrong in the higher dimensional scalar-tensor solutions of string theory which are non-singular, but contain p-branes, like the 3+1 dimensional membrane black hole of type IIB supergravity.
Very nice. Thankyou
January, 15, paper published on ArXiv, now reading....
What's the title?
@@trebabcock if you can't figure that out....
@@barefeg See, you expended the effort necessary to leave a comment, but instead of giving me the title you decide to insult me. Nice.
@@trebabcock why would I give you the answer? It's academia after all...
@@barefeg Because it's the nice thing to do? Academia is about collaborating and working with others. So that point did not make sense.
Let’s find the Einstein metric which must exist in a non compact kobayashi hyperbolic metric space
What is mi in a causal path? And what is tau?
affine parameters; could be anything when suitably chosen
Why diamond? Why not just cone?
you mean like 10:49? A cone would not work because “p” is not affected by the corners of a cone because they are spacelike to “p”. Remove the corners from your cone and you get a diamond
@@qz1771 Not that I ever aspired to understand this. I think IQ of 150 is min. required to fully understand.
@@qz1771 It seems in 3:15 he mentions future cone so there are two cones stacked onto each other, the future one being reversed (this I learned from Penrose video). So whatever comes to p must come via a cone and whatever leaves q must leave as a cone. The surface is light propagation and inside are speeds less than light speed. Since the surface is never-ending, they will intersect. But why is 2D diamond depicted and not 3D cones?
@@Burevestnik9M730 Hi Jovan, both a 2D diamond and 3D cones (arranged so that the top one is inverted) would work. A presentation can only display 2D images, however. Moreover, your description of 3D isn't completely faithful still since we are describing 4D processes. After some getting used to the 2D representation will get familiar!
@@qz1771 Light cone is 4D representation. Orthogonal is t axis and horizontal 2D representation of 3D space (Penrose).
Cool..
De ja vu !
Keith butler are you referring to them covering the subject many times or some personal case of seeing the same thing many times or is there a personal vein to this.
Keith butler oh wait I get it saw the comments before the video.