Metric Definition and 1000 Threads

METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model developed by the University of Idaho, that uses Landsat satellite data to compute and map evapotranspiration (ET). METRIC calculates ET as a residual of the surface energy balance, where ET is estimated by keeping account of total net short wave and long wave radiation at the vegetation or soil surface, the amount of heat conducted into soil, and the amount of heat convected into the air above the surface. The difference in these three terms represents the amount of energy absorbed during the conversion of liquid water to vapor, which is ET. METRIC expresses near-surface temperature gradients used in heat convection as indexed functions of radiometric surface temperature, thereby eliminating the need for absolutely accurate surface temperature and the need for air-temperature measurements.

The surface energy balance is internally calibrated using ground-based reference ET that is based on local weather or gridded weather data sets to reduce computational biases inherent to remote sensing-based energy balance. Slope and aspect functions and temperature lapsing are used for application to mountainous terrain. METRIC algorithms are designed for relatively routine application by trained engineers and other technical professionals who possess a familiarity with energy balance and basic radiation physics. The primary inputs for the model are short-wave and long-wave thermal images from a satellite e.g., Landsat and MODIS, a digital elevation model, and ground-based weather data measured within or near the area of interest. ET “maps” i.e., images via METRIC provide the means to quantify ET on a field-by-field basis in terms of both the rate and spatial distribution. The use of surface energy balance can detect reduced ET caused by water shortage.
In the decade since Idaho introduced METRIC, it has been adopted for use in Montana, California, New Mexico, Utah, Wyoming, Texas, Nebraska, Colorado, Nevada, and Oregon. The mapping method has enabled these states to negotiate Native American water rights; assess agriculture to urban water transfers; manage aquifer depletion, monitor water right compliance; and protect endangered species.

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  1. J

    What is the Meaning of r in Schwartzschild Metric?

    What is the meaning of r in the Schwartzschild metric?. ds^2 = \frac{{dr^2 }}{{1 - \frac{{2GM}}{{c^2 r}}}} + r^2 (d\theta ^2 + \sin ^2 \theta d\varphi ^2 ) - c^2 \left( {1 - \frac{{2GM}}{{c^2 r}}} \right)dt^2 If you were to actually measure the radius, your observation would be affected by...
  2. S

    Plugging Godel Metric's Line Element into Software for Covariant Einstein Tensor

    http://en.wikipedia.org/wiki/G%C3%B6del_metric Could someone please plug the line element for the Godel metric (seen on the above wiki page) into some software to see what comes out for the Einstein tensor in a coordinate basis (preferably the covariant version rather than the contravariant...
  3. Z

    Relative speed of orbiting rockets in Schwarzschild metric

    Homework Statement Two rockets are orbiting a Schwarzschild black hole of mass M, in a circular path at some location R in the equatorial plane θ=π/2. The first (rocket A) is orbiting with an angular velocity Ω=dΦ/dt and the second (rocket B) with -Ω (they orbit in opposite directions). Find...
  4. B

    Derivative of metric with respect to metric

    I'm hoping someone can clarify for me, I have seen the following used: \frac{\partial}{\partial g^{ab}}\left( g^{cd} \right) = \frac{1}{2} \left( \delta_a^c \delta_b^d + \delta_b^c \delta_a^d\right) I understand the two half terms are used to account for the symmetry of the metric tensor...
  5. binbagsss

    The Significance of FRW Metric for Cosmological Redshift

    So from a killing tensor the FRW metric is known to possess, for a massless particle we find the well known result that as the universe expands the frequency of the photons decreases . But , what does this do for gr ? Was this known to happen before gr ? Thanks a lot. (I know it is used to...
  6. T

    How to differentiate a term with respect to metric

    Homework Statement for example: ##\frac{\partial(F^{ab}F_{ab})}{ \partial g^{ab}} ## where F_{ab} is electromagnetic tensor. or ##\frac{\partial N_{a}}{\partial g^{ab}}## where ##N_{a}(x^{b}) ## is a vector field. Homework EquationsThe Attempt at a Solution i saw people write ##F^{ab}F_{ab}##...
  7. S

    Godel Metric Cosmological Constant

    What is the cosmological constant for the Godel metric?
  8. T

    Covariant and partial derivative of metric determinant

    Homework Statement is this statement is true : ##\nabla_\mu \nabla_\nu \sqrt{g} \phi = \partial_\mu \sqrt{g} \partial_\nu \phi## Homework EquationsThe Attempt at a Solution well we know ##\nabla_\mu \sqrt{g} =0## so it moves back : ## \nabla_\mu \sqrt{g} \nabla_\nu \phi =\sqrt{g} \nabla_\mu...
  9. T

    Varying the action with respect to metric

    Homework Statement i want to find the variation of this action with respect to ## g^{\mu\nu}## , where ##N_\mu(x^\nu)## is unit time like four velocity and ##\phi## is scalar field. ##I_{total}=I_{BD}+I_{N}## ## I_{BD}=\frac{1}{16\pi}\int dx^4\sqrt{g}\left\{\phi...
  10. binbagsss

    Does Metric Signature Affect Torsion Definition?

    I'm looking at 2 sources. One has it defined as ##T^{c}_{ab}=-\Gamma^{c}_{ab}+\Gamma^{c}_{ba}## And the other has ##T^{c}_{ab}=\Gamma^{c}_{ab}-\Gamma^{c}_{ba}## ##T## the torsion tensor and ##\Gamma^{c}_{ab}## the connection. Or is it more that different texts use different conventions? Thanks.
  11. U

    What Is the FRW Metric and How Is It Applied in General Relativity?

    Homework Statement (a) Find the FRW metric, equations and density parameter. Express the density parameter in terms of a and H. (b) Express density parameter as a function of a where density dominates and find values of w. (c) If curvature is negligible, what values must w be to prevent a...
  12. binbagsss

    Deriving Schwarz Metric Weak Limit: Carroll's Lecture Notes 1997

    I'm looking at Carroll's lecture notes 1997, intro to GR. Equation 7.27 which is that he's argued the S metric up to the form ##ds^{2}=-(1+\frac{\mu}{r})dt^{2}+(1+\frac{\mu}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}## And argues that we expect to recover the weak limit as ##r \to \infty##. So he then...
  13. S

    Type Godel Metric Line Element - Get Help Here!

    Can someone please type out the line element for the Godel metric (including any and all c terms and any other terms that one might omit if they were using natural units to set terms like c = 1)? I ask this because different sources on line have it written out in different ways which look...
  14. U

    What Are the Steps to Solve Einstein Equations for This Metric?

    Homework Statement [/B] (a) Find the christoffel symbols (b) Find the einstein equations (c) Find A and B (d) Comment on this metric Homework Equations \Gamma_{\alpha\beta}^\mu \frac{1}{2} g^{\mu v} \left( \partial_\alpha g_{\beta v} + \partial_\beta g_{\alpha v} - \partial_\mu g_{\alpha...
  15. U

    What Are the Steps to Solve This Space-Time Metric Homework?

    Homework Statement (a) Find ##\dot \phi##. (b) Find the geodesic equation in ##r##. (c) Find functions g,f,h. (d) Comment on the significance of the results. Homework Equations The metric components are: ##g_{00} = -c^2## ##g_{11} = \frac{r^2 + \alpha^2 cos^2 \theta}{r^2 +\alpha^2}##...
  16. K

    Metric tensor with diagonal components equal to zero

    Hello, Let ##g_{jk}## be a metric tensor; is it possible for some ##i## that ##g_{ii}=0##, i.e. one or more diagonal elements are equal to zero? What would be the geometrical/ topological meaning of this?
  17. L

    Proving Horizon Structure of Complicated Metric

    Hi there, I have a metric with g_{rr}=\frac{1}{r^2-2cr}. From this it is clear there exist coordinate singularities at r=0 and r=2c. I believe that the outer horizon is the event horizon and the inner horizon is a Cauchy horizon. However, I do not know what I need to do in order to prove that...
  18. C

    Sphere of Uniform Density: Exact Solution to Einstein's Field Eqns?

    Is an exact solution to Einstein's Field Equations known for the interior of a sphere of uniform density (to approximate a star or planet, for example?)
  19. D

    Twin Paradox in Kerr Metric - Help Needed

    Hi. I've been struggling with a formulation of the twin paradox in the Kerr metric. Imagine there are two twins at some radius in a Kerr metric. One performs equatorial circular motion whilst the other performs polar circular motion. They separate from one another and the parameters of the...
  20. T

    Observables vs. continuum and metric?

    Space in quantum mechanics seems to be modeled as a triplet of real numbers, i.e. a continuum. Same happens in special relativity. General relativity I do not know (nor field theories). And then we apply the Pythagorean theorem and triangle inequality and so forth... I have a few general...
  21. U

    What is the Geodesic Equation for FRW Metric's Time Component?

    Taken from Hobson's book: Metric is given by ds^2 = c^2 dt^2 - R^2(t) \left[ d\chi^2 + S^2(\chi) (d\theta^2 + sin^2\theta d\phi^2) \right] Thus, ##g_{00} = c^2, g_{11} = -R^2(t), g_{22} = -R^2(t) S^2(\chi), g_{33} = -R^2(t) S^2(\chi) sin^2 \theta##. Geodesic equation is given by: \dot...
  22. E

    Metric for uniform constant acceleration?

    What does the metric look like for constant uniform acceleration, say in the x-direction? ds^2 = g_tt (cdt)^2 + g_xx (dx)^2 g_tt = ? g_xx = ?
  23. J

    Equivalent Metrics From Clopen Sets

    Homework Statement Prove that if ##(X,d)## is a metric space and ##C## and ##X \setminus C## are nonempty clopen sets, then there is an equivalent metric ##\rho## on ##X## such that ##\forall a \in C, \quad \forall b \in X \setminus C, \quad \rho(a,b) \geq 1##. I know the term "clopen" is not a...
  24. binbagsss

    Tod & Hughston GR Intro: FRW Metric Derivation w/ R=6k or R=3k?

    I'm looking at Tod and Hughston Introduction to GR and writing the metric in the two forms; [1]##ds^{2}=dt^{2}-R^{2}(t)(\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2}))## [2] ##ds^{2}=dt^{2}-R^{2}(t)g_{ij}dx^{i}dx^{j}## where...
  25. binbagsss

    Einstein Hilbert action, why varies wrt metric tensor?

    The principle of least action states that the evolution of a physical system - how a system progresses from one state to another- is given by a stationary point of the action. So I think this is varying the path and keeping two points fixed- the points of the initial and final state I know...
  26. binbagsss

    Raising Indices with Minkowski Metric: Solving Weak Field Approx

    I have the expression ##g_{ab}=\eta_{ab}+\epsilon h_{ab}##, The indices on ##h^{ab}## are raised with ##\eta^{ab}## to give ##g^{ab}=\eta^{ab}-\epsilon h^{ab}## I am not seeing where the minus sign comes from. So I know ##\eta^{ab}\eta_{bc}=\delta^{a}_{c}## and...
  27. Y

    Help with the variation of the Ricci tensor to the metric

    I should calculate the variation of the Ricci scalar to the metric ##\delta R/\delta g^{\mu\nu}##. According to ##\delta R=R_{\mu\nu}\delta g^{\mu\nu}+g^{\mu\nu}\delta R_{\mu\nu}##, ##\delta R_{\mu\nu}## should be calculated. I have referred to the wiki page...
  28. C

    Stress tensor from action in Landau-Ginzburg field theory

    I would appreciate any help with the following question: I know that for relativistic field theories, the stress tensor can be obtained from the classical action by differentiating with respect to the metric, as is explained on the wikipedia page...
  29. binbagsss

    Stationary/ Static Conditions Metric?

    What are the sufficient / necessary conditions for a metric to be stationary / static? - If the metric components are independent of time in some coordinate system , is this sufficient for stationary? - I've read for static if a time-like killing vector is orthogonal to a family of...
  30. K

    What is Rieman for a conformal metric?

    Hi... The ordinary plain vanilla conformal metric in spherical coordinates is: ds2 = a(t)2[dt2 - dr2/(1 - kr2) - r2 (dΘ2 + sin2(Θ) d(φ)2)] where a(t) is a function of time only. I am trying to find out what Rieman, Ricci and the Scalar Curvature are for this common metric when k=1 and...
  31. N

    Wave equation given a cosmological inflationary metric

    Hi everybody! Can you explain me how I can obtain wave equation given a metric? For example, if I have this metric $$g_{μν}=diag(−e^{2a(t)},e^{2b(t)},e^{2b(t)},e^{2b(t)})$$, how can derive the relation $$\frac{1}{\sqrt{g}}\partial _t(g^{00}\sqrt{g}\partial _t...
  32. Muthumanimaran

    Metric Transformations: Explained with Diagrams

    When I study about the transformation of coordinates, especially while defining gradient, curl, divergence and other vector integral theorem in different co-ordinate system, a concept called metric is defined and it is said to used for transform these operators in different co-ordinates, it is...
  33. W

    MHB Real Analysis Help: Metric Spaces

    Show that two metrics p and T on the same set X are equivalent if and only if there is a c > 0 such that for all u,v belong to X, (1/c)T(u,v)=<p(u,v)=<cT(u,v) Please help me , I'm so confused about Real Analysis.
  34. K

    Question about Metric Tensor: Can g_{rr} be Functions of Coordinate Variables?

    Hello Say, the metric tensor is diagonal, ##g=\mbox{diag}(g_{11}, g_{22},...,g_{NN})##. The (null) geodesic equations are ##\frac{d}{ds}(2g_{ri} \frac{dx^{i}}{ds})-\frac{\partial g_{jk}}{\partial x^{r}}\frac{dx^{j}}{ds}\frac{dx^{k}}{ds} = 0## These are ##N## equations containing ##N## partial...
  35. M

    Questions about FLRW Metric: Finiteness, Radial Coord & More

    Hello, I have two questions regarding the FLRW metric, it is more about its interpretation. The metric reads: ##dl²=dt²-a²(\frac{dR²}{1-kR²}+R²d\Omega²)## where ##a## is the radius of the 3-sphere (universe), and ##R=r/a## a normalized radial coord. What I don't understand is this statement...
  36. E

    Metric matrix for binary star system?

    What does the metric matrix look like for a binary star system? Does each follow its usual geodesic about the other? It seems like the solution would have to be different somehow than that for a tiny planet circling a big sun.
  37. F

    Angular velocity calculation from Schwarzschild metric

    Hello, I need to find the angular velocity using Schwarzschild metric. At first I wrote the metric in general form and omitted the co-latitude: ds2=T*dt2+R*dr2+Φ*dφ2 and wrote a Lagrangian over t variable: L = √(T+R*(dr/dt)2+Φ*(dφ/dt)2) now I can use the Euler–Lagrange equations for φ...
  38. binbagsss

    FRW Metric: Parameter k & Space/Space-Time Relationships

    This is probably a stupid question but does k=1,0,-1 correspond to closed,flat,open refer to space or space-times? Looking at a derivation what each geometrically represents is only done when talking about the spatial part of the FRW metric. As space can be flat and space-time still curved...
  39. D

    Elliptical Orbits In The Schwarzschild Metric

    I was just wondering how you would go about calculating the proper time for an observer following a freely falling elliptical orbit in a Schwarzschild metric. I am happy with how to calculate the proper time for a circular orbit and was wondering whether if you had two observers start and end...
  40. T

    Solving Einstein's Field Equation: The Schwarschild Metric & Beyond

    What is the next easiest solution to einstein field equation after the schwarschild metric?
  41. binbagsss

    Understanding Homogeneity & Isotropy in FRW Metric

    So in deriving the metric, the space-time can be foliated by homogenous and isotropic spacelike slices. And the metric must take the form: ##ds^{2}=-dt^{2}+a^{2}(t)\gamma_{ij}(u)du^{i}du^{j}##, where ## \gamma_{ij} ## is the metric of a spacelike slice at a constant t QUESTION: So I've read...
  42. binbagsss

    Deriving FRW Metric: Ricci Vector Algebra Explained

    I'm looking at: http://arxiv.org/pdf/gr-qc/9712019.pdf, deriving the FRW metric, and I don't fully understand how the Ricci Vectors eq 8.5 can be attained from 7.16, by setting ##\partial_{0} \beta ## and ##\alpha=0## I see that any christoffel symbol with a ##0## vanish and so so do any...
  43. binbagsss

    When can a metric be put in diagonal form?

    I'm looking at deriving the Schwarzschild metric in 'Lecture Notes on General Relativity, Sean M. Carroll, 1997' and the comment under eq. 7.8, where he seeks a diagnoal form of the metric... - Is it always possible to put a metric in diagonal form or are certain symmetries required? - What...
  44. binbagsss

    General Relativity: Manifold/Sub-Manifold Metric Theorem Q-Schwarzschild

    I'm looking at Lecture Notes on General Relativity, Sean M. Carroll, 1997. I don't understand eq 7.4 from the theorem 7.2. As I understand, theorem 7.2 is used when you have submanifold that foilate the manifold, and the submanifold must be maximally symmetric. I know that 2-spheres are...
  45. U

    Proper distance, Area and Volume given a Metric

    Homework Statement [/B] (a) Find the proper distance (b) Find the proper area (c) Find the proper volume (d) Find the four-volume Homework EquationsThe Attempt at a Solution Part (a) Letting ##d\theta = dt = d\phi = 0##: \Delta s = \int_0^R \left( 1-Ar^2 \right) dr = R \left(1 -...
  46. ChrisVer

    FRW Metric in d Dimensions: Can I Expand?

    I was wondering if I can expand the FRW metric in d spatial dimensions, like: g_{\mu \nu}^{frw} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & - \frac{a^2(t)}{1-kr^2} & 0 & 0 \\ 0 & 0 & - a^2(t) r^2 & 0 \\ 0 & 0 & 0 & -a^2 (t) r^2 \sin^2 \theta \end{pmatrix} \rightarrow g_{MN} = \begin{pmatrix} g_{\mu...
  47. binbagsss

    Schwarzchild metric spherically symmetric space or s-t?

    This is probably a stupid question, but, is the Schwarzschild metric spherically symmetric just with respect to space or space-time? Looking at the derivation, my thoughts are that it is just wrt space because the derivation is use of 3 space-like Killing vectors , these describe 2-spheres, and...
  48. T

    The Schwarzschild Metric: Obtaining Equation M=Gm/c^2 & Newton Law at Infinity

    How do you obtain this equation M=Gm/c^2. What does M stand for? Is is Newton law at infinity? Again what is this Newton law at infinity?
  49. Math Amateur

    MHB Metric Spaces - Fixed Point Theorem (Apostol, Theorem 4.48)

    I need help with the proof of the Fixed Point Theorem for a metric space (S,d) (Apostol Theorem 4.48) The Fixed Point Theorem and its proof read as follows: In the above proof Apostol writes: " ... ... Using the triangle inequality we find for m \gt n, d(p_m, p_n) \le \sum_{k=n}^{m-1}...
  50. L

    Kerr metric and rotating stars

    I have recently come across the notion that Kerr metric describes the spacetime outside a rotating black hole but not outside a rotating (electrically neutral) star. Unlike Schwarzschild metric, which works both for non-rotating spherically symetric black hole without charge as well as any other...
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