□h=-16πT

Philosophy isn't too bad. So far we've covered Ancient Greek philosophy (Plato, Aristotle and Socrates), Judeo-Christian concept of God and Meta-/Normative-Ethics, that's the first module. At the moment we're doing Kantian ethics (Catagorical Imperative) and religious philosophy. It starts off a bit tedious but looks interesting for the rest of AS and A2. Student room?

Maths, further maths, physics and philosophy. I took philosophy because I hate biology and chemistry and there wasn't anything else that really complements my first three options besides those three. That and I like it.

AS, it's boring. Not that A2 is any better. Oh well, my teacher lets me read stuff of my own in lesson. How about yourself? What exams do you have?

AS, it's boring. Not that A2 is any better.

Easy, really. Saying that, I'll fail it. Still have S1, D1 and philosophy exams left to sit next week. God, I hate A level physics.

This is a question from the AQA (I think) Physics module 1 paper isn't it? I sat that the other week. That's pretty much what I put for a) and b). I mentioned something about work being done by the electron in leaving the atom, there being an increase in potential energy and decrease in kinetic as it escapes.

If you want to be accurate about these things, it's for a spherically symmetric static body. For rotating and non-symmetric bodies you have to use different equations. For a rotating, charged, axially symmetric body, there are two "horizons": the ergosphere and an event horizon similar to that of a Schwarzschild black hole.

Nevermind...

A body under the influence of the field of some other body doesn't feel the effect of the force vector pointing in the opposite direction, so they don't cancel out.

I've missed the square on the mass term in the Lagrangian, by the way. Should be [math]{\cal L}=\tfrac{1}{2}\left[\left(\partial_{\mu}\phi\right)^2-m^2\phi^2\right][/math] And I missed the imaginary term from the commutation relations. That should be [math][\phi(\vec{x}), \pi(\vec{y})]=i(2\pi )^3\delta^{(3)}(\vec{x}-\vec{y})[/math] Conjugate momentum should be the derivative of the Lagrangian with respect to the time derivative of the field. Sorry about those few mistakes, bit of a lapse in concentration there.

No, the formula for the Schwarzschild radius involves rest mass. Gravitation is frame invariant, if you speed up your field doesn't increase, relativistic mass isn't real.

I have of course used the Schrodinger interpretation (no time depedance of opertors), but one can easily use the Heisenberg interpretation (time dependance), by performing a time evolution on the above and using a couple of useful identities satisfied by the creation and anihilation operators, to yield the same results.

I'll show you part of a simple method of quantisation. The Lagrangian of a free real Klein-Gordon field is [math]{\cal L}=\tfrac{1}{2}\left[\left(\partial_{\mu}\phi\right)^2-m^2\phi\right][/math] Define the momentum conjugate to [math]\phi[/math] as [math]\pi(\vec{x})=\frac{\partial\phi}{\partial\dot{\phi}}[/math] Note here that [math]\phi[/math] is still a classical field. In passing from the discrete definition of the Hamiltonian (involving a summation) to the continuous case we have [math]H=\int d^3x\left[\pi\dot{\phi}-{\cal L}\right]=\int d^3x\tfrac{1}{2}\left[\pi^2+\left(\vec{\nabla}\phi\right)^2+m^2\phi^2\right][/math] Then taking the field and conjugate momenta to be operators we can enforce a commutation relation analogous to the regular commutation relations between position and momentum in the continuous case [math][\phi(\vec{x}), \pi(\vec{y})]=(2\pi )^3\delta^{(3)}(\vec{x}-\vec{y})[/math] If we then model the Klein-Gordon field as a harmonic oscillator, drawing analogy from the quantisation of a simple harmonic oscillator with a single frequency, we can expand [math]\phi(\vec{x})[/math] and [math]\pi(\vec{x})[/math] in terms of "ladder operators" [math]a_{\vec{p}}[/math] and [math]a_{\vec{p}}^{\dag}[/math] as a fourier transform with each [math]a_{\vec{p}}[/math] etc. corresponding to a particular Fourier mode [math]\vec{p}[/math]. [math]\phi(\vec{x})=\int\frac{d^3p}{(2\pi )^3}\frac{1}{\sqrt{2E_{\vec{p}}}}\left(a_{\vec{p}}e^{i\vec{p}\cdot\vec{x}}+a_{\vec{p}}^{\dag}e^{-i\vec{p}\cdot\vec{x}}\right)[/math] [math]\pi(\vec{x})=-i\int\frac{d^3p}{(2\pi )^3}\sqrt{\frac{E_{\vec{p}}}{2}}\left(a_{\vec{p}}e^{i\vec{p}\cdot\vec{x}}-a_{\vec{p}}^{\dag}e^{-i\vec{p}\cdot\vec{x}}\right)[/math] Where [math]E_{\vec{p}}=\sqrt{|\vec{p}|^2+m^2}[/math] is the energy of that particular mode. The commutation relations satisfied by the ladder operators are [math][a_{\vec{p}}, a_{\vec{q}}^{\dag}]=(2\pi )^3\delta^{(3)}(\vec{x}-\vec{y})[/math] Then calculating the Hamiltonian using this fourier expansion we get [math]H=\int\frac{d^3p}{(2\pi )^3}E_{\vec{p}}\left(a_{\vec{p}}a_{\vec{p}}^{\dag}+\tfrac{1}{2}[a_{\vec{p}}, a_{\vec{p}}^{\dag}]\right)[/math] The integral of the second term is an infinite constant (as it's proportional to [math]\delta(0)[/math]) and arises as the sum over all modes of the zero point energy [math]\frac{E_{\vec{p}}}{2}[/math]. The spectrum of the first term is easily quantised using the commutation relation satisfied between H and the respective ladder operators. The state [math]|0\rangle[/math], defined by [math]a_{\vec{p}}|0\rangle=0[/math] [math]\forall\vec{p}[/math], is known as the ground state, or vacuum, of the theory and is the state with the least energy. Each eigenstate of H with momentum [math]\vec{p}+\cdots +\vec{q}[/math] can be constructed from [math]|0\rangle[/math] with a creation operator [math]a_{\vec{p}}^{\dag}[/math] by [math]a_{\vec{p}}^{\dag}\cdots a_{\vec{q}}^{\dag}|0\rangle[/math]. The Hermitian conjugate of the creation operator (i.e. [math]a_{\vec{p}}[/math]) is an anihilation operator that lowers the eingenstate, [math]a_{\vec{p}}a_{\vec{q}}^{\dag}|0\rangle=|0\rangle[/math]. Therefore we have seen that the operator [math]a_{\vec{p}}^{\dag}[/math], and thus [math]\phi[/math], is responsible for the creation of discrete amounts of energy that we can identify with a "particle", with further reasoning (quotation marks because the phenomenon is not localised in the usual sense of a particle). More things can be said about this field, but I've only given it as a simple example of second quantisation. It's much more difficult to quantise other field theories, especially when interactions are involved, but the idea was to give you a taste of what it involves.

In it own reference frame we'd have [math]d\tau ^2=0[/math] The norm of its four-velocity would then be [math]\vec{U}\cdot \vec{U}=\frac{d\vec{x}}{d\tau}\cdot\frac{d\vec{x}}{d\tau}=\frac{0}{0}[/math] The rest frame of light is thus undefined.

In relativity we use greek indices. You've used latin indices in your expression for the Einstien tensor etc. It's no biggy, I just thought I'd let you know in case you didn't already. Latin indices are usually used for spinor indices. Oh and the Lagrangian I gave above is for the vacuum. To couple gravity to matter you just add a Lagrangian for matter and include a normalisation constant [math]\frac{1}{8\pi}[/math] (units in which G=c=1).

The rest frame of light is not definable. Negative speed, no, as speed is the magnitude of velocity, a vector, and is thus positive. Velocity can be negative, if one defines what is meant by positive velocity.

As an inertial frame O approaches the event horizon, time runs as usual. However from an external reference frame the time it takes for O to reach the horizon is infinite.

For example, quantising the Klein-Gordon field theory using the second quantisation method: you get yourself some commutation (or anticommutation as in the Dirac field) relations for the field and cannonical momentum and expand them as a fourier integral in terms of "ladder operators". Calculate the Hamiltonian using that expansion and find its eigenvalue spectrum.

You need to quantise your theory, otherwise it's just a classical theory.

Circumference of circle [math]2\pi r[/math] Infinitesimal area of circle [math]2\pi rdr[/math] Integrate over this between 0 and R (the radius of the circle) [math]\int _0^R2\pi rdr=\pi R^2[/math] Similarly with a sphere.

That's an analogy. It's easier than writing [math]G^{\alpha \beta}=8\pi T^{\alpha \beta}[/math] on the board and confusing everyone.

Frames freely falling in a gravitational field are inertial. If you were going to try and create a Newtonian analogue of GR you'd have to include the weak equivalence principle, which would remove the acceleration from your equations. Einstein arrived at the idea of inertial motion over a curved space-time being gravity because of this equivalence principle.

The functional integral formalism allows for simple computation of amplitudes/correlation functions by use of [math]\int {\cal D}\psi e^{\lbrack i\int d^4x{\cal L}\rbrack}[/math] Where the functional integral [math]\int {\cal D}\psi [/math] runs over each configuration of the field [math]\psi [/math] (with Lagrangian density [math]{\cal L}[/math]) between the end points.

There are a few ways of defining the Riemann tensor, all of which are equivalent. The first is the term arising in the commutator of covariant derivatives [math]\lbrack \nabla_{\alpha}, \nabla_{\beta} \rbrack V^{\nu} =R^{\nu}_{\mu\alpha\beta}V^{\mu}[/math] (Summation running over repeated indices) This is for a torsion free Lorentzian connection, where the connection coefficient (Christoffel connection [of the second kind] in GR) is symmetric in its contravariant indices. For manifolds with torsion there is another term subtracted from the end. Another is as follows: parrallel transport a vector about a closed loop and upon it returning to its initial position the difference of the two defines the loss of parrallelism, i.e. curvature. The other can be seen from the equation of geodesic deviation. The gist of it is that it is a quantitative measure of the curvature of a manifold. One can define the Riemann tensor for non-coordinate bases using the Weyl curvature tensor.

The Ricci tensor is a contraction of the Riemann curvature tensor on its first and third indices, i.e. [math]R_{\alpha\beta}=\Sigma_{\nu}R^{\nu}_{\alpha\nu\beta}[/math] and because of the contraction it is symmetric. The Ricci scalar is the only contraction one can form from just the Riemann tensor, the contraction of any of the other indices is simply the Ricci tensor with various sign changes. The Ricci scalar is the simplest scalar one can form from the Riemann tensor. It is a further contraction of the Riemann tensor [math]R=\Sigma_{\alpha, \beta}g^{\alpha\beta}R_{\alpha\beta}=\Sigma_{\nu, \alpha, \mu, \beta}R^{\nu\alpha\mu\beta}R_{\nu\alpha\mu\beta}[/math] The Einstein tensor is a symmetric divergence free tensor that essentially describes the curvature of the manifold under consideration. The divergence free characteristic follows from the Bianchi identies and demonstrates energy-momentum conservation since [math] G^{\alpha\beta}=kT^{\alpha\beta}[/math] [math]\nabla _{\beta}G^{\alpha\beta}=k\nabla _{\beta}T^{\alpha\beta}=0[/math] [math]k\nabla _{\beta}T^{\alpha\beta}=0 [/math] Thus energy-momentum is conserved. The derivation of the field equations originally given by Einstien appeals to such nature of the Einstein tensor as constructed from the Ricci tensor and scalar: It being second order in the metric, as the Newtonian field equation is second order in the potential. The Einstein tensor is second order in the metric as it is constructed from Ricci tensors which in turn are constructed from the Riemann tensor, which consists of terms involving first derivatives and quadratic terms in the Christoffel connection (in GR any way) that is constructed from first order terms in the metric. Symmetric, thus in the equality of G with T both sides are symmetric in their indices. G embodies the curvature, thus formalising Einstein's idea of gravity being the result of inertial motion along a curved space-time. Energy-momentum conservation by virture of the Bianchi identities. The rigorous and formal alternative to this intuitive derivation implements the principle of least action on the Einstein-Hilbert Lagrangian density [math]{\cal L}=R\surd(-det[g_{\alpha\beta}])[/math] This is acreditable to Hilbert, who collaborated with Einstein during the development of GR.