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-rw-r--r--math/tensors68
-rw-r--r--physics/LIGO48
-rwxr-xr-xphysics/fig/cross-section-diagram.jpgbin0 -> 1871788 bytes
-rw-r--r--physics/general relativity17
-rw-r--r--physics/gravitational waves56
-rw-r--r--reStructured Text2
-rw-r--r--sheets/fig/biohazard.png (renamed from fig/biohazard.png)bin179 -> 179 bytes
-rw-r--r--sheets/rst2
8 files changed, 177 insertions, 16 deletions
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
new file mode 100755
index 0000000..13fdca5
--- /dev/null
+++ b/physics/fig/cross-section-diagram.jpg
Binary files differ
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
index ae4629d..ae4629d 100644
--- a/fig/biohazard.png
+++ b/sheets/fig/biohazard.png
Binary files differ
diff --git a/sheets/rst b/sheets/rst
index e105cba..4714fe3 100644
--- a/sheets/rst
+++ b/sheets/rst
@@ -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