□h=-16πT

Thanks again for your answer and time !

Still not confortable with that' date=' but I must live with it if I want to go deeper in my learning of SR.[/quote']

 

If we're dealing with a theory in which c plays a fundamental role, why not absorb this constant into the background of the theory itsellf, so to speak, by simply redefining the units? If you're not comfortable with it, just think of it as a mathematical simplification, we're just being pragmatic: it gets to be a pain in the arse in long calculations, in any branch of physics , if we have to keep writing out the same constant that really doesn't affect the manner or progression of the calculation itself, except until we want a numerical value, where we can just easily convert back to SI. In GR geometrised units are used, where G=c=1, and in quantum physics its often convention to take units where h-bar=1 or h-bar=c=1.

You've got

[math](9-x^2)^{\tfrac{1}{2}}[/math]

Then you use the chain rule.

And around a blackhole the shortest route always leads straight into a blackhole... that is what a blackhole does, warp space-time so much that it will bend in on itself (ie. into the blackhole).

No it doesn't. You can get stable orbits about black holes.

Your derivative is wrong by the way.

[math]y=-\sqrt{9-x^2} [/math]

[math]y'=-\frac{1}{2}(9-x^2)^{-\frac{1}{2}}\frac{d(-x^2)}{dx}=\frac{x}{\sqrt{9-x^2}}[/math]

I can't remember how you do coefficient of friction and all the gubbins, but the problem without coefficients of friction just requires that the kid's kinetic energy at the bottom of the pipe be the same as its potential energy at its initial position, height=3. Can you not integrate friction coefficients into conservation of energy.

Well for a weak field we can consider the metric as a small perturbation from flat space-time metric

[math]g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}[/math]

Where the field h is not necessarilly a tensor, but a function added to each component of the Lorentzian metric. However, if we consider Minkowski space-time as a background, the field h will transform as a (0 2) tensor, so we can consider it as a tensor in such circumstances. If we now give ourselves an expression for the Einstein tensor in terms of our perturbed flat metric we can impose certain simplifying restrictions on h. These are the traceless-transverse gauge and the Lorentz (or de Donder) gauge, the latter being similar to that in classical electrodynamics. If we do this we end up with the weak field version of the field equations of general relativity which serves the basis of linearised field theory (linearised because we only one the terms linear in h)

[math]\left(\frac{1}{c^2}\frac{\partial}{\partial t}-\nabla^2\right)h_{\alpha\beta}=-16\pi T_{\alpha\beta}[/math]

Where T is the usual symmetric stress energy tensor of the generating field.

"A First Course In General Relativity"-B. Schutz has a good discussion on the treatment of gravitational waves in linearised field theory, including its detection and the energy radiated from bodies by their gravitational wave emission. Note that spherically symmetric bodies do not radiate gravitational radiation, as their quadrupole and higher moments vanish due to the symmetry.

Jesus, he polluted myspace with this as well.

You trying to put me out of a job?

Officially, no. Off the recod, yes.

no - that is a mistake. I will have to think about it some more.

Danke, I wish it did work though . I hate functional integrals, as elegant as the whole procedure is for deriving Feynman rules.

Or have you done something that gets rid of that factorial?

Cheers, man.

and the time evolutions opreators?

It allows one to take some state defined at some particular time and find the state at a further point in time. So one can remove time dependance, which can then be replaced by evolution operators, and makes things a bit easier in places.

The Hamiltonian is one of the most important elementary quantities in quantum mechanics. It arises because of time, and gives the energy of the system, in a round about way.

The more we learn about QM the more time seems to be Obsolete.

Oh, really? Yeah, I guess you're right, after all Hamiltonians and time evolutions opreators play a minor, insignificant role.

I was trying to approach it, as the question suggested, by constructing it out of Wilson lines and using a functional integral.

There should be factorials in the exponent that you've missed out, which will muck up the end result.

*shameless bump*

Anyone?

Angular momentum is conserved for central systems without external forces. That this results in angular momentum conservation arises from the following relationship between turning moment G and angular momentum J in an analogous manner to the conservation of linear momentum from Newton's second law

[math]\vec{G}=\frac{d\vec{J}}{dt}[/math]

Hey, I've got this problem from Peskin & Schroeder (chapter 15). I'm not particularly confident with functional integration, as I'm pretty new to it, and working through such a book by myself is pretty tricky in places. Well here goes

The Wilson Loop for QED is defined as

[math]U_p(z, z)=\exp \left[-ie\oint_pdx^{\mu}A_{\mu}\right][/math]

With the Wilson line defined similarly (just change it so that there's not a closed contour integral and with the end points (z,z) changed to (z, y), or whatever you like).

Where A is the photon field, the gauge connection asociated with transformations in U(1).

Now it says: using functional integration, show that the expectation of the Wilson loop for the electromagnetic field free of fermions is

[math]\langle U_p(z, z)\rangle =\exp \left[-e^2\oint_pdx^{\mu}\oint_pdy^{\nu}\frac{g_{\mu\nu}}{8\pi^2(x-y)^2}\right][/math]

Where x and y are integrated over the closed loop P.

I think the Feynman propogator might be useful here, so to save anyone having to look it up,

[math]D_F^{\mu\nu}(x-y)=\int\frac{d^4q}{(2\pi )^4}\frac{-ig^{\mu\nu}e^{-ip\cdot (x-y)}}{q^2+i\epsilon }[/math]

(The imaginary term in the denominator of the integrand is the application of the Feynman boundary conditions, ensuring the convergence of the Gaussian integral involved in the derivation of the propogator.)

I have a vague idea of how to go about it, but I'm not particularly confident about it, it's finding the relevant starting point that's causing me problems, i.e. putting together and computing the functional integral for the expectation. I'm just going through this to gain some confidence in functional integration etc. so if anyone can give a few pointers as to going about this it'd be much appreciated. This isn't a homework question, if that puts anyone off helping me, I doubt my A level teacher would set something like this .

Cheers

and yet Einstien still forgot to account for space time into his equations and did realise . it was teh "biggest mistake f his life" lol.

Cosmological constant

The Bohr model only works for hydrogen, as in its derivation it uses a Coloumb potential between two point particles: a proton and an electron. It fails to account for the spin of the electron and proton, as the derivation is non-relativistic, which results in a slight, almost unoticable, discrepency between the predicted emission spectra and the actual emission spectra, known as the lamb shift.

The conserved current associated with a translation symmetry

Buggery, beaten to it.

It's looks bad with inline TeX' date=' but that's the general FLRW metric. The [imath']\bar{r}[/imath] is defined as:

 

[math]\bar{r} =\begin{cases} R \sinh(r/R), &\mbox{globally hyperbolic} \\ r, &\mbox{globally flat} \\R \sin(r/R), &\mbox{globally spherical} \end{cases}[/math]

 

...where [imath]R[/imath] is the radius of curvature.

I'm not familiar with the bar notation, so I either didn't see it or neglected it.

Thanks patcalhoun and h=-16nT

If I understand right then a metric is an equation to calculate the distance between 2 locations or more precisely' date=' since it is spacetime, between to events.

I see that time is not subject to the scale factor, so time doesn't expand like space. Why since we are suppose to calculate spacetime and time is a dimesion like the others of space ?

I have a question that is bugging me since I started studying Relativity, why use ct for the time dimension instead of just t ?

Thanks[/quote']

 

Because ct has units of length in SI, which makes the metric dimensionally consistent. The convention is to adopt units in which c=G=1 in S/GR, to make things simpler.

Thanks Tycho

I followed the link and found that they also talk about "space stretching" :

 

I followed the link to the Robertson-Walker metric and found that equation:

[math]ds^2 = c^2 dt^2-a(t)^2[dr^2+\bar{r}^2 d\Omega^2][/math]

 

Can you explain it a little bit' date=' in layman term ?

Thanks again[/quote']

 

That's the Robertson-Walker metric for a globally flat space-time. The general metric is

 

[math]ds^2=c^2dt^2-R(t)^2\left[\frac{dr^2}{1-kr^2}+r^2d\Omega^2\right][/math]

 

The k can be +1, 0 or -1, as the radial coordinate r can always be rescaled so that k takes on one of these values. k is the value of Ricci scalar, which is a constant becasue space-time is modelled as being homogeneous and isotropic. k=1 is positive curvature, and the resulting model is a closed spherical topology. k=0 we have zero curvature, a globally flat model. k=-1 we have negative curvature, and the resulting model is open and hyperbolic.

 

The equations governing the actual expansion can be derived from energy-momentum conservation.

That was Galileo's principle of relativity, not Einstein's. Einstein used this to show the invariance of c from Maxwell's equations governing the electromagnetic interaction.

It is non-sensical to use a photon rest frame to conduct any observations. This is simply because such a rest frame is undefined, as the four-velocity is indeterminate (0/0) etc.

I'm pretty sure Hermann Minkowski (one of Einsteins teachers) came up with the idea of unifying space and time...which helped general relativity to be realised. The idea stemmed from Lorentz geometry.

Yeah, I think he did. I can't be bothered to get the 1905 paper by Einstein out and check, but I'm pretty sure H. Minkowski introduced the space-time interval (hence the Minkowski interval) and the geometry of SR in 1906 (or '07).

The main bulk of "On The Electrodynamics of Moving Bodies" explained the Michelson-Morely experiment concisely and postulating the Lorentz invariance of c supported by a mathematical derivation from the Maxwell equations.

Who was the first one to propose that time was a fourth dimension that complimented the three dimensions of space, thereby establishing the concept of spacetime. The name "Lorenz" comes to mind. Is this right?

Introducing time as an additional property of the universe together with space by itself is nothing special and does not constitute a definition of space-time really. The important result is the invariance of the space-time interval, by themselves spatial and temporal intervals are relative to the observer, whereas an interval in space-time is frame independant. I don't know if you already knew this.

Lorenz is the name of a prominant guy in weather models and non-linear dynamics (chaos theory). Lorentz is the guy you're after; he was the dude that, subsequent to the MM experiment and prior to the '05 paper on SR, proposed length contraction and time dilation to explain the results. However his idea was seen merely as a mathematical trick to explain the "negative" results. He conducted other work into the subject as well.