Gauge freedom is the freedom of choosing a coordinate system. Fixing a gauge means choosing a particular coordinate system.
Gauge tranformation is Lie derivative along some arbitary vector here.
Line element
\begin{eqnarray} \tilde g _ {00} &=& -a^2(1+2 A Y) \newline \tilde g _ {0j} &=& -a^2 B Y _ j \newline \tilde g _ {ij} &=& a^2(\gamma _ {ij} +2 H _ L Y \gamma _ {ij} +2 H _ T Y _ {ij} ) \end{eqnarray}
Energy momentum tensor is
\begin{eqnarray} \tilde T^0 _ {\phantom{0}0} = -\rho (1+\delta Y) \newline \tilde T^0 _ {\phantom{0} j} = (\rho + p)(v - B) Y \newline \tilde T^j _ {\phantom{j}0 } = -(\rho + p)v Y^{j} \end{eqnarray}
For a infinitesimal gauge transformation along some vector (X=T\partial _ t + L^i \partial _ i), gauge variables are
Symbol | Physics | Gauge Transformation | Note |
---|---|---|---|
$\tilde A $ |
Through that we can find out gauge invariant variables
]]>This post is aimed to make clear what frame are we in.
In general relativity, we often transform coordinates. Here is an example.
The general form of metric with spherical space component is
\begin{equation} \mathrm ds^2 = - \gamma(r,t)c^2\mathrm dt^2 + \beta(r,t)c\mathrm dr\mathrm dt + \alpha(r,t)[\mathrm dr^2 + r^2 (\mathrm d\theta^2 + \sin^2\theta \mathrm d\phi^2)]\label{MetricForm1} \end{equation}
With a transformation $\alpha(r,t)r^2 = r’^2$,
Then compose the integral multiplier
And finally,
In general
\begin{equation} \mathrm ds^2 = -b(r,t)c^2\mathrm dt^2 + a(r,t)\mathrm dr^2 + r^2(\mathrm d\theta^2 + \sin^2\theta\mathrm d\phi^2) \label{MetricForm2} \end{equation}
Then what? The two forms of metric demonstrate different properties. Take Birkhoff theorem as an example. The results could be very different startting from the form (\ref{MetricForm1}) and (\ref{MetricForm2}).
It is obviously very important to show what the coordinate transformation means and what frame are the observers in indicated by the coordinates.
The first point is for the invariance of frames/coordinates. The second point is for the success of Newtonian’s theory on our earth.
Why do we believe the third point? The answer is that we don’t have to. Here we propose it is because the simplicity of such quasilinear equations, i.e., We have a bunch of theorems on this system, including its existance of solutions, Couchy problem, wave propagation etc.
We can use both 1&2 and 1&3 to derive Einstein’s equation. That is 2 and 3 are identical when 1 is considered.
This is an application of stationary principal and Hilbert action or Hilbert action plus a $\Lambda$.
The only possible second-order Euler-Lagrange expression obtainable in a four dimensional space from a scalar density of the form $\scr L = \scr L(g_{\mu\nu})$ is
Thus modification could be
All spherically symmetric solutions of Einstein’s equations in vacuum must be static and asymptotically flat, without $\Lambda$
Actually, this can be extended to a $\Lambda$ space only keeping the static result.
The generic final state of gravitational collapse is a Kerr-Newman black hole, fully specified by its mass, angular momentum and charge
Also, “in the context of General Relativity with a cosmological constant all expanding universe solutions should evolve towards de Sitter space.”[^{1}] This is only valid in some situation.
R. M. Wald. Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant. Phys. Rev. D, 28(8):2118–2120, Oct 1983.↩
f(R) gravity background equations and perturbation equations:
https://raw.github.com/CosmologyTaskForce/ModiGraviDoc/master/ModifiedGravityDoc.nb
background equations and perturbation equations
https://raw.github.com/CosmologyTaskForce/ModiGraviDoc/master/ModifiedGravityDoc.nb
]]>The test of gravity theories can be viewed as test of the fundations of gravity theories and the the theories themselves, say test of equivalent principle and general relativity or f(R) gravity theory. Thus we should break down general relativity theory into several stages. Here, we use the following table to do so.
Theory | Mach | WEP | EEP | SEP | GC | Notes |
---|---|---|---|---|---|---|
GR | Partial | Y | Y | Y | Y |
Theory | Topoplogy | Manifold | Connection | Metric |
---|---|---|---|---|
GR | No torsion | Non-metricity tensor vanishes |
Theory | Gravitational Waves | Newtonian Limit | GR Limit | Notes |
---|---|---|---|---|
GR |
Most items in mathematics are the same in different theories.
In Newtonian system, the acceleration of an object will be \[ \vec a \propto \frac{\vec F}{m_I} \]
In a static and uniform gravitation field, the gravity force is \[ \vec G = - g m_G \hat r \]
Thus the acceleration in this case should be \[ \vec a \propto -\hat r g \frac{m_G}{m_I} \]
When $m_G/m_I$ is constant, the falling accerelation are the same for different objects with same mass. However, if $m_G/m_I$ is not a constant, say $m_G\ne m_I$, different objects would fall at different acceleration.
Now if we put two ball with different mass on the Eotvos torsion balance, the balance would rotate and we can measure it.
Detection of $R^k_{0l0}=(1/c^2)\partial^2\Phi/\partial x^k\partial x^l \sim 10^{-32} \text{cm}^{-2}$.
Anisotropy of gravitation/electromagnetism is not proved in our galaxy.
Similar to Eddington and Dyson’s bending light observation, radio signals serve as a more precise experiment to test Einstein’s theory. And these experiments are against scalar tensor theories because scalar tensor theories give a smaller bending angle (1.66 second of arc less than the observations).
Tables constructed according to arXiv:1106.2476v3.
Test of fundamental principles
Experiment | Results | Note | |
---|---|---|---|
WEP | Eotvos torsion balance | $\eta = (0.3 \pm 1.8) \times 10^{-13}$ | More precise in space exp. [[^1]a] [^{1}] [^{2}] |
Gravitational redshift of light | [^{3}] | ||
EEP | Hughes-Drever Experiment | $n \le 10^{-27}$ | [^{4}] [^{5}] |
Test of GR:
Experiment | Results | Note | |
---|---|---|---|
Null geodesics test | photon trajectory, spatial deflection | $\theta = (0.99992\pm 0.00023)\times 1.75’’$, where 1.75 is the theoretical value | Achieved through observing star position, etc [^{6}] |
Shapiro time-delay effect | $\Delta t = (1.00001\pm 0.00001)\Delta t_{GR}$ | [^{7}] [^{8}] | |
Time like geodesics | Anomalous perihelion precession | Just use the PPN formalism [^{9}] [^{10}] [^{11}] | |
Nordtvedt effect | $\eta = (-1.0 \pm 1.4) \times 10^{-*13}$ | [^{12}] [^{13}] | |
Spinning objects obiting | [^{14}] [^{15}] | ||
Small-range | Potential probing | [^{16}] [^{17}] | |
Radiation | Speed of gravitational waves | ||
Polarity of gravitational radiation | |||
Dynamics of source objects |
Eotvos experiment: using torsion balance to test the equality of gravitational mass and inertial mass. Wikipedia has a photo of how this works. ↩
$\eta=2\frac{ABS(a1-a2)}{ABS(a1+a2)}$. $a1$ and $a2$ are the accelerations of the two bodies in Eotvos torsion balance. Thus $\eta$ is the accleration difference of the two objects.↩
References: R. W. P. Drever. A search for anisotropy of inertial mass using a free precession technique. Philosophical Magazin, 6:683-687, May 1961. ; V. W. Hughes, H. G. Robinson, and V. Beltran-Lopez. Upper Limit for the Anisotropy of Inertial Mass from Nuclear Resonance Experiments. Physical Review Letters, 4:342-344, Apr. 1960. ; S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab, and E. N. Fortson. New limits on spatial anisotropy from optically-pumped 201 Hg and 199 Hg. Physical Review Letters, 57:3125–3128, Dec. 1986. ; T. E. Chupp, R. J. Hoare, R. A. Loveman, E. R. Oteiza, J. M. Richardson, M. E. Wagshul, and A. K. Thompson. Results of a new test of local Lorentz invariance: A search for mass anisotropy in 21 Ne. Physical Review Letters, 63:1541–1545, Oct. 1989.↩
Hughes-Drever Experiment: test the isotropy of mass and space through the NMR spectrum, or the mono-metric spacetime.↩
S. S. Shapiro, J. L. Davis, D. E. Lebach, and J. S. Gregory. Measurement of the Solar Gravitational Deflection of Radio Waves using Geodetic Very-Long-Baseline Interferometry Data, 1979 1999. Physical Review Letters, 92(12):121101, Mar. 2004.↩
References, I. I. Shapiro. Fourth Test of General Relativity. Physical Review Letters, 13:789–791, Dec. 1964. ; B. Bertotti, L. Iess, and P. Tortora. A test of general relativity using radio links with the Cassini spacecraft. Nature, 425:374–376, Sept. 2003.↩
Shapiro time-delay effect: time delay when light travels through a massive object.↩
Observational data for the value of perihelion precession of Mercury are summarized in E. V. Pitjeva. Modern Numerical Ephemerides of Planets and the Importance of Ranging Observations for Their Creation. Celestial Mechanics and Dynamical Astronomy, 80:249–271, July 2001. ↩
PPN formalism is the lowest order of GR.↩
Anomalous precession:↩
K. Nordtvedt. Equivalence Principle for Massive Bodies. I. Phenomenology. Physical Review, 169:1014–1016, May 1968. ; J. G. Williams, S. G. Turyshev, and D. H. Boggs. Progress in Lunar Laser Ranging Tests of Relativistic Gravity. Physical Review Letters, 93(26):261101, Dec. 2004, arXiv:gr-qc/0411113.↩
Nordtvedt effect: massive objects in Eotvos torsion balance experiments. We can use the whole Earth-Moon system to test this effect.↩
There is a Lense Thirring effect here. GPB has done this.↩
GR can be reduced to Newtonian potential at small range.↩
Currently, most of the modification has a Yukawa potential form.↩
CfA Redshift Survey, the Center for Astrophysics Redshift Survey, is the first redshift survey.
Measure radial velocities for all galaxies brighter than 14.5 and high galactic latitude in the merged catalogs of Zwicky and Nilson (UGC). {Completed}
Distribution of the galaxies in the northern celestial hemisphere with an apparent blue magnitude of 15.5 inside redshift 15,000 km/s,
In this map, Red V < 3000 km/s; Blue 3000 < V < 6000 km/s; Magenta 6000 < V < 9000 km/s; Cyan 9000 < V < 12000 km/s; Green 12000 < V km/s.
2dFGRS, 2dF Galaxy Redshift Survey, is a major spectroscopic survey (integrated with the 2dF QSO survey).
Objectives: {Completed}
Parameters:
Results
Galaxy distribution results from completed survey:
A rendered result of 2dF Galaxy Survey data
Links:
2dF QSO Redshift Survey, 2QZ, is already integrated into 2dFGRS.
Objectives: {Completed}
Parameters:
Objectives: {Completed}
2MASS, the Two Micron All Sky Survey, “is designed to close the gap between our current technical capability and our knowledge of the near-infrared sky. In addition to providing a context for the interpretation of results obtained at infrared and other wavelengths.” It shows the large-scale structure of the Milky Way and the Local universe.
Objectives: {Completed}
Parameters:
Results
Links:
2MRS, the 2MASS Redshift Survey, aims to map the distribution of galaxies and dark matter in the local universe, based on galaxy selection of 2MASS.
Objectives: 45000 galaxies up to K=11.75mag {Completed}
Parameters:
Results
All sky map of local universe in infrared
Links:
VLA FIRST Survey: Faint Images of the Radio Sky at Twenty-cm using NRAO Vary Large Array.
Objectives: “to produce the radio equivalent of the Palomar Observatory Sky Survey over 10,000 square degrees of the North and South Galactic Caps” {Completed}
Parameters:
Results
Links:
EDisCS: ESO Distant Clusters Survey
Objectives: {Completed}
Parameters:
Results
Links:
LCRS, the Las Campanas Redshift Survey
Objectives: “provide a large galaxy sample which permits detailed and accurate analyses of the properties of galaxies in the local universe” {Completed}
Parameters:
Results
Two dimensional representations of the redshift distribution
We can see voids and walls.
Links:
Results
Cone diagram
Links:
CNOC: the Canadian Network for Observational Cosmology
Objectives: {Completed}
Parameters:
Results
Determine Omega
Links:
Objectives: {Completed}
Parameters:
Results
Links:
SDSS: Sloan Digital Sky Survey
There are 3 programs until now. SDSS is too famous and too complicated to list here. Use the links to read it on SDSS website.
Links:
GAMA: Galaxy And Mass Assembly Survey
Results
Three animations to illustrate some of the results, (blue for 10^8 to 10^9 solar masses; green for 10^9 to 10^10 solar masses; orange for 10^10 to 10^11 solar masses; red for more than 10^11 solar masses)
Links:
UKIDSS: UKIRT Infrared Deep Sky Survey, is a successor to 2MASS and made up of five surveys and includes two deep extra-Galactic elements, one covering 35 square degrees to K=21 and the other reaching K=23 over 0.77 square degrees.
Objectives:
Parameters:
Results
Links:
Pan-STARRS: the Panoramic Survey Telescope & Rapid Response System
Objectives: (Originally) detect potentially hazardous objects in the solar system, and other solar system astronomy and cosmology.
Parameters:
Results
Links:
PRIMUS: PRIsm MUlti-object Survey, is the largest faint galaxy spectroscopic redshift survey to date.
Objectives:
Parameters:
Results
Links:
BOSS: Baryon Oscillation Spectroscopic Survey. FUTURE SURVEY
Objectives:
Parameters:
Results
Links
Objectives: “designed to probe the origin of the accelerating universe and help uncover the nature of dark energy by measuring the 14-billion-year history of cosmic expansion with high precision”
Parameters:
Results
Links
LSST: the Large Synoptic Survey Telescope
Objectives:
Parameters:
Results
Links
Template for surveys:
Objectives: XXXXXX {Completed}
Parameters:
Results
Links: