okay so so this time we are going to talk about some new topic so we are going to talk about so-called ADM them okay no activity so we so we have finished the discussion about some specific solutions Einstein equation so we are just jump out from easiest of solutions back to firmly locked so here this ADF onanism is a Hamiltonian from Needham of gravity of general activity and here this a DM name it's a three three ninety three the name of workout this formalism look how the Hamiltonian of general activity okay so how to do and this Hamiltonian romnesia so I mean as I am sure all of you have learned classical mechanics you know in classical mechanics you can write down the Lagrangian and then using this Lagrangian and define the conjugate momentum corresponding to the configuration variable and then you write your Lagrangian in terms of those configuration variable and momentum P and Q and and then extract the Hamiltonian of the system from from your account yeah okay and here is a game is it's just the same it's followed basically the same scheme but we apply this Lagrangian for for general activity of gravity and okay so here our action of gravity is so called ina Huebel action we have discussed volume integral on the four dimensional manifold and so as I discussed as we discussed before and so this is the Box turn of action and you have to add a boundary term to make it well-defined especially to make the variational principle well-defined you have to add another term which is people Gibbons 14 boundary and the boundary this is before and then you use this action you perform the variational principle zero and these give you the Einstein equation this is something we we did okay so now what we want to do is to extract Hamiltonian and extract a conjugate momentum variable and configuration variable from from this action and also extract a Hamiltonian from this action so here now what we want to do we want we want to write this Lagrangian the following form is going to have have these signs and hoop action to be this kind of type right is as a PQ dot - a Hamiltonian formalism so so once we can we are able to write in this form so a good thing about it is is that in the end of these analysis we can reformulate Einstein equation in terms of Hamilton's equation that Einstein equation can be cast into Hamilton's equation namely Q dot equal to partial H ro and P dot - we want to write n sign question in this kind of a good thing I'm still learning in classical mechanics you know these kind of Hamilton's equations nice because it transforms a second-order differential equation into first order differential equation and now of course you get more variables you get momentum variable in addition to Q ok so here we learn Einstein equation is the second order differential equation so here basically this kind of analysis helped you to write a second or a second order differential equation - first order differential equation and it's so we can formulate this system in terms of initial value problem ok so we are we can simply find the initial values and our initial conditions and the initial condition is just a point of phase space and the phase space is the space of P and Q s that's in classical mechanics and the initial condition is just a point in phase space and then the time evolution following the hamon equation is just that should that give you a trajectory in your face pace starting from the initial point okay so so here some key points is that so I'm the reason why we need is Hamilton Hamilton's equation or Hamiltonian formulation of gravity is to port and so firstly we need it because we we want we want any initial value formulation of gravity so we need a initial value formulation okay and this is very useful especially nowadays in numerical relativity and starting rotation weights and what one to do is put this Hamilton's equation port for gravity in in computer and specify the initial values and let this equation grow computational computationally convenient and and secondly we these Hamilton's what Hamiltonian formulation is necessary to do the quantization needed because we know that quantum mechanics what quantum mechanics does is is to quantize the Hamiltonian formalism of a particle next and so considering an article we have to have a Hamiltonian and then we quantize this Hamiltonian and plug in the Schrodinger equation right so definitely before the quantization we have to have a Hamiltonian so that's why we need support bond to quantize gravity we have to firstly find the Hamiltonian point little o of gravity okay so these are two motivations right so one observation let me put so there is one observation observation is that so Hamiltonian formula so so Hamiltonian so to do the Hamiltonian formalism we need we need to specify the time a time let's call this time T so so here the reason that programmer in because we want to write is and he would actually into this P Q dot minus H so here there is something called Q dot yeah and this Q dot definitely depend on the choice of time okay so which means to in order to define this coming on your own rhythm we need to firstly specify a specific time right so this means that we need a so called three plus one decomposition of space-time okay so remember that I mean before we talk about we talk about activity we say that I'm general relativity is the point of general relativity is coordinate independent coordinate transformation is a gauge transformation of gravity so are all physics is that depend our choice of coordinate but in this case and we are going to break this gauge invariant a little bit in order to develop this Hamiltonian for it so this is a special special ality of of general relativity or of gravities although the coordinate transformation is is a transformation but in order to develop this Hamiltonian formalism we have to sort out fix that at least fix the time coordinate we have to fix at time T so once we fix the coordinate of course we'll break this case and this case freedom but we sort of we have to do that for or how come in order to develop the okay so here what is the 3+1 decomposition so firstly we have to assume the background manifold structure M equals are both sick ok and here this R is the time flow and Sigma is our space so here the 3 + 1 combination for the background manifold is that we have to decompose the many body into a time direction and space direction and the dimension of space is 3 ok so we always comes there for dimensional activity so the dimension of C minus 3 so I'm in in figure so we consider this to plot is more respective yeah so these are all sides so this is called a foliation of space-time so this is variation variation of the space-time it means that consider that space-time is a made by spatial slices lots of special spice slices heard about these special slices is the parameter parameter is by by T being the time and so the T coordinate axis is like I mean it's kind of arbitrary following from the past to the future so you view this this is actually I'm drawing it's tangent vector at call ta yeah so this let me say let me put this the first three toes according Li see this is the cordon line and and then then at every point there's a tangent vector this tangent vector is called TA okay so this TA is just a partial party in our notation okay and this is so here what I'm going is a spacetime and we consider this space-time has a matrix G a B and this manifold is its audio and and then because we have the spatial slice we have we have the time direction so this ta can be decomposed by by using the bases by using at least the normal vector of of the spatial slice so because once we have the hypersurface Sigma T so we can we know we have a normal vector yeah we have the metric we have the hypersurface we have a normal vector and therefore we can we can some decompose is ta by the component certain components of na and also certain component projected on D and the spatial slack okay so here this ta ta can be written in terms of a linear combination of na with coefficient what I call and ya plus and they and this na is a projection of ta onto the spatial slide okay and this n which is hot laps function is called n is a function so everything is the vector field all the quantity in this equation our vector field so it depends on space and time yeah so what I'm drawing here is just a vector field located at one point the vector so so this end is called lapse function out lapse function and these na is called shipped back ship divided alright so here this na is tangent tangent to see okay this 10 is an AM and here na and a equals to minus 1 yeah so here we and this na is a normalized normal vector alright so now I suppose with we consider a coordinate system one let's call it the T X 1 X 2 X 3 this is so here this T is definitely the time coordinate and and this X 1 X 2 3 is coordinate and Sigma