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diff --git a/math/tensors b/math/tensors new file mode 100644 index 0000000..8fda6a5 --- /dev/null +++ b/math/tensors @@ -0,0 +1,68 @@ +============================================ +Tensors, Differential Geometry, Manifolds +============================================ + +.. note:: Most of this content is based on a 2002 Caltech course taught by + Kip Thorn [PH237]_ + + +On a manifold, only "short" vectors exist. Longer vectors are in a space tangent to the manifold. + +There are points (P), separation vectors (\Delta \vector P), curves ( Q(\zeta) ), tangent vectors ( \delta P / \delta \zeta \equiv \lim_{\Delta \zeta \rightarrow 0} \frac{ \vector{ Q(\zeta+\delta \zeta) - Q(\zeta) } }{\delta \zeta} ) + +Coordinates: \Chi^\alpha (P), where \alpha = 0,1,2,3; Q(\Chi_0, \Chi_1, ...) + there is an isomorphism between points and coordinates + +Coordinate basis: \vector{e_\alpha} \equiv \left( \frac{\partial Q}{\partial \Chi^\alpha} \right) + for instance, on a sphere with angles \omega, \phi: + \vector{e_\phi} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_\theta + +Components of a vector: + \vector{A} = \frac{\partial P}{\partial \Chi^\alpha } + +Directional Derivatives: consider a scalar function defined on a manifold \Psi(P) + \partial_\vector{A} \Psi = A^\alpha \frac{\partial \Psi}{\partial \Chi^\alpha} + + ???????? + +Mathematicians like to say that the coordinate bases are actually directional derivatives + +Tensors +------------ + +A tensor T has a number of slots (say 3) and takes a vector in each slot and returns a real number. It is linear in vectors. + +\epic{T} ( \alpha \vector{A} + \beta \vector{B}, \vector{C}, \vector{D}) = + \alpha \epic{T} (\vector{A}, \vector{C}, \vector{D}) + + \beta \epic{T} (\vector{B}, \vector{C}, \vector{D}) + +The number of "slots" is the rank of the tensor. + +Even a regular vector is a tensor: pass it a second vector and take the dot +product to get a real. + +Define the metric tensor g(\vector{A}, \vector{B}) = \vector{A} \dot \vector{B} + +Inner Product: + \Delta P \dot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 + A \dot B = 1/4[ (A+B)^2 - (A-B)^2 ] + +Tensor Product: + ???????????????? + +Spacetime +-------------- + +Two types of vectors. + +Timelike: \vector{\Delta P} + (\vector{\Delta P})^2 = -(\Delta \Tau)^2 + +Spacelike: \vector{\Delta Q} + (\vector{\Delta Q})^2 = +(\Delta S)^2 + +Because product of "up" and "down" basis vectors must be a positive Kronecker +delta, and timelikes squared come out negative, the time "up" basis must be +negative of the time "down" basis vector. + + diff --git a/physics/LIGO b/physics/LIGO new file mode 100644 index 0000000..23fba64 --- /dev/null +++ b/physics/LIGO @@ -0,0 +1,48 @@ +======================================================================= +LIGO: Laser Interferometer Gravitational Observatory +======================================================================= + +.. warning:: This is a rough work in progress!! Likely to be factual errors, poor grammer, etc. + +.. note:: Most of this content is based on a 2002 Caltech course taught by + Kip Thorn [PH237]_ + +Noise Sources +~~~~~~~~~~~~~~~~~~~~~~ + +For initial LIGO, seismic noise dominates below about 60Hz, suspension thermal +noise between 60 and 180Hz, and radiation pressure shot noise above 180Hz. + +Sensitivity +~~~~~~~~~~~~~~~~~~~~~ + +Advanced LIGO will use 40kg sapphire test masses with sensitivity of about 10e-19 meters: 1/10000 of an atomic nucleus, 10e-13 of a wavelength, and half of the entire mirror's wave function. + +LISA +~~~~~~~~~~~~~ + +5e6 km separations between three spacecraft, 1 (astronomical unit, ~1.5e8 from the sun. 1 watt lasers. +The heterodyne detection is of the beat frequencies at each spacecraft of the +two incoming beams. Doppler shifts of spacecraft must be taken into account, +due not only to sun radiation pressure etc, but varying gravitational fields +from planetary orbits. + +The test masses inside LISA should be free falling and have relative +separations stable to 10e-9 cm (10e-5 wavelength of light). + +LISA's sensitivity is in the milihertz regime. + +.. note: (insert LISA noise curve?) + +Data Analysis +~~~~~~~~~~~~~~~~~~~~ + +Using matched filtering (eg, take cross correlation between two waveforms, +integrating their product), frequency sensitivity will be around the inverse +of the number of cycles of waveform (for LIGO, around 20,000 cycles, for LISA +around 200,000 cycles). + +This technique requires known, theoretically derived waveforms (within +phase/amplitude). There are other methods when we don't have good guesses +about the waveform we are looking for... + diff --git a/physics/fig/cross-section-diagram.jpg b/physics/fig/cross-section-diagram.jpg Binary files differnew file mode 100755 index 0000000..13fdca5 --- /dev/null +++ b/physics/fig/cross-section-diagram.jpg diff --git a/physics/general relativity b/physics/general relativity new file mode 100644 index 0000000..9c2a90b --- /dev/null +++ b/physics/general relativity @@ -0,0 +1,17 @@ +=========================== +General Relativity +=========================== + +.. warning:: This is a rough work in progress!! Likely to be factual errors, + poor grammer, etc. + +.. note:: Most of this content is based on a 2002 Caltech course taught by + Kip Thorn [PH237]_ + +*See also `math/tensors </k/math/tensors>`_* + + +References +---------------- + +.. [PH237] `Gravitational Waves`:title: (aka ph237), a course taught by Kip Thorne at Caltech in 2002. See http://elmer.tapir.caltech.edu/ph237/ for notes and lecture videos. diff --git a/physics/gravitational waves b/physics/gravitational waves index db2e667..81080c3 100644 --- a/physics/gravitational waves +++ b/physics/gravitational waves @@ -2,14 +2,14 @@ Gravitational Waves ======================= -:Author: bnewbold@mit.edu +.. warning:: This is a rough work in progress!! Likely to be factual errors, poor grammer, etc. .. note:: Most of this content is based on a 2002 Caltech course taught by Kip Thorn [PH237]_ Raw Info ----------------- -Rank 4 Riemann tensors, will cover different gages. +Rank 4 Riemann tensors, will cover different gauge. Waves are double integrals of curvature tensor... @@ -20,38 +20,66 @@ Invariance angles: (Spin of quantum particle) = :latex:`$2 pi$` / (invariance an Graviton has :latex:`$\pi$` invariance angle, so it is spin 2; photons have unique :latex:`$\arrow{E}$` vector, so invariance angle is :latex:`$2\pi$`, spin 1 -Also describes spin by the group of lorentz transformations which effect propogation. +Also describes spin by the group of Lorentz transformations which effect propagation. -Two polarizations: cross and plus, corresponding to spin of particles aligning wiht or against propagation? (Ref: eugene vickner? reviews of modern physics) +Two polarizations: cross and plus, corresponding to spin of particles aligning with or against propagation? (Ref: Eugene Vickner? reviews of modern physics) -Waves' multipole order $\geq$ spin of quantum = 2 for graviton ((??)) +Waves' multipole order :latex:`$\geq$` spin of quantum = 2 for graviton ((??)) -Waves don't propogate like E, because mass monopoles don't oscillate like charges. +Waves don't propagate like E, because mass monopoles don't oscillate like charges. -:latex:`$ h \req \frac{G}{c^2} \frac{M_0}{r} + \frac{G}{c^3} \frac{M'_1}{r} + \frac{G}{c^4} \frac{M''_2}{r} + \frac{G}{c^4} \frac{S'_1}{r} + \frac{G}{c^5} \frac{S''_1}{r}$` +:latex:`$ h \approx \frac{G}{c^2} \frac{M_0}{r} + \frac{G}{c^3} \frac{M'_1}{r} + \frac{G}{c^4} \frac{M''_2}{r} + \frac{G}{c^4} \frac{S'_1}{r} + \frac{G}{c^5} \frac{S''_1}{r}$` First term: mass can't oscillate Second term: momentum can't oscillate -Third term: mass qudrupole moment dominates +Third term: mass quadrupole moment dominates Fourth term: angular momentum can't oscillate Fifth term: current quadrupole Energy ---------------- -Quick calculation: for a source with mass M, size L, period P, the quadupole moment $M_2 \req M L^2$, h \req 1/c^2 (newtonian potential energy) ???? +Quick calculation: for a source with mass M, size L, period P, the quadrupole moment $M_2 \approx M L^2$, h \approx 1/c^2 (Newtonian potential energy) ???? h on the order of $10^{-22}$ -Propogation +Propagation ----------------- -When wavelength much less than curvature of universe (background), then gravitational waves propagate like light waves: undergo red shifts, gravitational lensing, inflationary redshift, etc. +When wavelength much less than curvature of universe (background), then gravitational waves propagate like light waves: undergo red shifts, gravitational lensing, inflationary red shift, etc. + +Sources +------------- + +Inspirals of bodies into super-massive black holes + Eg, white dwarfs, neutron stars, small black holes. + Super-massive black holes are expected near the centers of galaxies. + Low frequencies (LISA); waveforms could hold data about spacetime curvature + local to the black hole. + Waveforms could be very difficult to predict. + +Binary black hole mergers + Broadband signals depending on masses. + +Neutron Star/Black hole mergers + Stellar mass objects existing in the main bodies of galaxies. + Higher frequencies (LIGO and AdvLIGO). + +Neutron Star/Neutron Star mergers + Have actual examples in our galaxy of these events; but final inspiral rate + is so low that we have must listen in other galaxies. Merger waves will + probably be lost in higher frequency noise, so can't probe local + gravitational curvature. + May observe "tails" of waves: scattering off of high curvature around the + binary. + +Pulsars (spinning neutron stars) + Known to exist in our galaxy. Spectrum ---------------- High Frequency: Above 1 Hz, LIGO (10 Hz to 1kHz), resonant bars - Small black holes (2 to 1k suns), neutron stars, supernovae + Small black holes (2 to 1k suns), neutron stars, supernovas Low frequency: 1Hz and lower, LISA (10^-4 Hz to 0.1 Hz), Doppler tracking of spacecraft Massive black holes (300 to 30 million suns), binary stars @@ -63,12 +91,12 @@ Extreme Low Frequency: 10^-16 Hz, Cosmic Microwave Background anisotropy Detectors ----------------- -$\Delta L = h L \lreq 4 \times 10^{-16} \text{cm}$ +:m:`$\Delta L = h L ~ \leq 4 \times 10^{-16} \text{cm}$` LIGO (10 Hz to 1kHz) Also GEO, VIRGO, TAMA (?), AIGO -LISA (10^-4 Hz to 0.1 Hz) +LISA (10e-4 Hz to 0.1 Hz) Resonant Bars ~~~~~~~~~~~~~~~ diff --git a/reStructured Text b/reStructured Text index 62ecf2c..c624ded 100644 --- a/reStructured Text +++ b/reStructured Text @@ -15,5 +15,5 @@ I also have a cheatsheet_. .. _docutils: http://docutils.sourceforge.net/ .. _Git Wiki: /k/gitwiki/ -.. _cheatsheet: /k/restructuredtextcheatsheet/ +.. _cheatsheet: /k/sheets/rst/ diff --git a/fig/biohazard.png b/sheets/fig/biohazard.png Binary files differindex ae4629d..ae4629d 100644 --- a/fig/biohazard.png +++ b/sheets/fig/biohazard.png @@ -44,7 +44,7 @@ Option list -o at least 2 spaces between option & description ================ ============================================================ ================ ============================================================ -Explicit Markup Examples (visible in the `text source <cheatsheet.txt>`_) +Explicit Markup Examples (visible in the `text source <raw/>`_) ================ ============================================================ Footnote .. [1] Manually numbered or [#] auto-numbered (even [#labelled]) or [*] auto-symbol |
