What is Metric: Definition and 1000 Discussions

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. George Keeling

    I Linearized Gravity & Metric Perturbation when Indices Raised

    I have just met linearized gravity where we decompose the metric into a flat Minkowski plus a small perturbation$$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},\ \ \left|h_{\mu\nu}\ll1\right|$$from which we 'immediately' obtain $$g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}$$I don't obtain that. In my rule book...
  2. facenian

    I A problem of completeness of a metric space

    Hi, I found this problem in Munkres' topology book, and it seems to be contradictory: Let X be a metric space. (a) Suppose that for some ϵ>0, every ϵ-Ball in X has compact closure. Show that X is complete. (b) Suppose that for each x\in X there is an \epsilon>0 such as the ball B(x,\epsilon) has...
  3. ergospherical

    I Proving Lorentz Metric on Real Type (1,0;1,0) Tensors in Wald Ch. 13

    In ch. 13, pg.349 of Wald it's asked to prove that ##g_{AA'BB'} = \epsilon_{AB} \bar{\epsilon}_{A'B'}## is a Lorentz metric on ##V## (containing the real elements of the vector space ##Y## of type ##(1,0;1,0)## tensors). Given the basis ##t^{AA'} = \dfrac{1}{\sqrt{2}}(o^A \bar{o}^{A'} + \iota^A...
  4. A

    I Showing Determinant of Metric Tensor is a Tensor Density

    I'm trying to show that the determinant ##g \equiv \det(g_{ij})## of the metric tensor is a tensor density. Therefore, in order to do that, I need to show that the determinant of the metric tensor in the new basis, ##g'##, would be given by...
  5. A

    I Expressing Vectors of Dual Basis w/Metric Tensor

    I'm trying to understand why it is possible to express vectors ##\mathbf{e}^i## of the dual basis in terms of the vectors ##\mathbf{e}_j## of the original basis through the dual metric tensor ##g^{ij}##, and vice versa, in these ways: ##\mathbf{e}^i=g^{ij}\mathbf{e}_j##...
  6. Safinaz

    A Deriving an action from a metric

    Hello! The paper I study is related to string theory and modified gravity theories topics. As they say in page 5 “The four-dimensional effective theory now follows by substituting Eq. (13) into the original action, Eq. (4)” I wonder how did they drive a 4- dimensional effective metric...
  7. Haorong Wu

    I Geodesics in Schwarzschild metric

    Hello, there. I am learning the chapter, The Schwarzschild Solution, in Spacetime and geometry by Caroll. I could not grasp the idea of circular orbits. It starts from the equations for ##r##, $$\frac 1 2 (\frac {dr}{d \lambda})^2 +V(r) =\mathcal E$$ where $$V(r)=\frac {L^2}{2r^2}-\frac...
  8. Charles_Xu

    I Schwarzschild Metric Singularity: Why?

    Why does the Schwarzschild metric have a singularity at r=0 if it is only valid outside the spherically symmetric static mass?
  9. Paige_Turner

    B What's the distance metric for a compactified dimension?

    I asked that question on another forum here. The 2 answers I got before the question was closed by an angry mod said: You wouldn't understand the answer. Don't ask that question. Ask about a Riemann sphere instead. You're too lazy to look up the answer in [a GR textbook that I don't own]...
  10. LCSphysicist

    How Do You Compute Killing Vectors for a Given Metric?

    I want to find all the killing vectors of the metric ##x²dx² + xdy²##. We could guess somethings by intuition and check it, but i decided to use the equation itself. Unfortunatelly, i realized that i am not sure how to manipulate the equation $$L_{\chi}g_{ab} = g_{ad}\partial_{b}...
  11. nomadreid

    I Shouldn't this definition of a metric include a square root?

    In https://mathworld.wolfram.com/InnerProduct.html, it states "Every inner product space is a metric space. The metric is given by g(v,w)= <v-w,v-w>." In https://en.wikipedia.org/wiki/Inner_product_space , on the other hand, "As for every normed vector space, an inner product space is a metric...
  12. ergospherical

    I Left Invariant Metric: What I Don't Understand

    I haven't learned about Lie Groups yet, but came across this question. What I don't understand: - is the semi-direct product ##R_+ \ltimes R^4## here a matrix group with elements ##\begin{pmatrix} \lambda & x^{\mu} \\ 0 & 1 \end{pmatrix}##? And is the group multiplication then matrix...
  13. Dukon

    A Kerr Metric Time Dilation Formula: Deriving Absolute Form

    Just as the time dilation formula for the Schwarzschild metric in terms of the position ##r## away from center of mass for a gravitational body and the Schwarzschild radius ##r_s = {2GM}/{c^2}## is given by $$ \tau = t \sqrt{1 - \frac{r_s}{r} } $$ so I'd like to know the corresponding...
  14. Haorong Wu

    I Metric: Proper Space vs Coordinate Space

    I imagine there is a isotropic space. Well, I would call it the proper space which will remain unchanged in any cases. And there is another space I call the coordinate space which will be distorted by gravitational field, i.e., metric. a) Suppose there are two stationary points. Their...
  15. Arman777

    A Understading geodesic equation/code for (+1,-1,-1,-1) Sch. Metric

    Hey guys, as you may remember, I have posted a question about plotting the orbit of timelike and null-like particles for a given metric. I think this discussion might be helpful for me and some other people in future studies. I have found an article, and in that article, the authors are using...
  16. Arman777

    Python Packages to Calculate orbits in Schwarzschild Metric

    I am looking for a Python Code/Package to calculate the orbits of the time-like and null-like particles in Schwarzschild metric (in spherical coordinates) Does anyone know such package ? Note: I am mostly looking for packages to calculate the RIGHT side of the given images (i.e the orbits...
  17. Arman777

    A Obtaining the Orbit equation (Effective Potential) from the Newtonian metric

    We can write the Newtonian metric in the form of $$ds^2 = -(1 - 2M/r)dt^2 + (1+2M/r)[dr^2 + r^2d\Omega^2]$$ In order to obtain the orbit equation I have written the constant of motion, $$e = (1 - 2M/r)(\frac{dt}{d\tau})$$ and $$l = r^2sin^2(\theta)(\frac{d\phi}{d\tau})$$ I can divide the...
  18. Arman777

    A Deriving Essential Quantities from Metric Tensor for GR Calculations

    I am working on a computational project about General Relativity. In this process, I want to code 'the stuff' that can be derivable from the metric tensor. So far, I have coded Riemann Tensor, Weyl Tensor, Einstein Tensors, Ricci Tensor, Ricci scalar. What are the other essential/needed...
  19. Arman777

    Wave equation for Schwarzschild metric

    I am trying to find the $$\nabla_{\mu}\nabla^{\mu} \Phi$$ for $$ds^2 = (1 - \frac{2M}{r})dt^2 + (1 - \frac{2M}{r})^{-1}dr^2 + r^2d\Omega^2$$ I have did some calculations by using $$\nabla_{\mu}\nabla^{\mu}\Phi = \frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu \nu}\partial_{\nu}\Phi)$$...
  20. A

    Divergence in Spherical Coordinate System by Metric Tensor

    The result equation doesn't fit with the familiar divergence form that are usually used in electrodynamics. I want to know the reason why I was wrong. My professor says about transformation of components. But I cannot close to answer by using this hint, because I don't have any idea about "x"...
  21. C

    A Induced Metric for Riemann Hypersurface in Euclidean Signature

    We know in Lorentzian signature spacetime, in the case of timelike or spacelike hypersurfaces ##\Sigma## with \begin{align} n^\alpha n_\alpha=\epsilon=\pm1 \end{align} where ##\epsilon=1## for timelike and ##-1## for spacelike. We can define a tensor ## h_{\alpha\beta}## on ##\Sigma## by...
  22. J

    How to calculate the four-momentum of a photon in FRW Metric

    I have calculated the Christoffel symbols for the above given metric, but I don't understand how to calculate a photon's four-momentum using this information. I believe it has something to do with the null geodesic equation but I can't understand how to put that information into the problem...
  23. J

    I Commutation between covariant derivative and metric

    First, we shall mention that it is known that the covariant derivative of the metric vanishes, i.e ##\nabla_i g_{mn} = 0##. Now I want tro prove the following: $$ \nabla_i A_k = g_{kn}\nabla_i A^n$$ The demonstration I encounter takes advantage of the Leibniz rule: $$ \nabla_i A_k = \nabla_i...
  24. Lilian Sa

    Diagonalizing a metric by a coordinate transformation

    I posted a thread yesterday and I think that I did not formulated it properly. So I have a metric ##{ds}^{2}=-{dt}^{2}+{dx}^{2}+2{a}^2(t)dxdy+{dz}^{2}## I was asked to find the the coordinate transformation so that I can get a diagonalized metric. so what I've done is I assumed a coordinate...
  25. Lilian Sa

    Diagonalizing a metric by a coordinate transformation

    hey there :) So I had a homework, and I was asked to diagonalize the metric ##{ds}^2=-{dt}^2+{dx}^2+2a^2(t)dxdy+{dz}^2## and to find the coordinate transformation for the coordinates of the new metric. so I found the coordinate transformation but the lecturer said that what I found is a...
  26. J

    I Metric Transformation b/w Inertial Frames: Analyzing Effects

    The metric tensor in an inertial frame is ## \eta = diag(-1, 1)##. Where I amb dealing with only 1-D space. The metric tranformation rule after a crtain coordinate chane is the following: $$ g_{\mu \nu} = \frac{\partial x^\alpha}{\partial x'^{\mu }} \frac{\partial x^\beta}{\partial x'\nu }...
  27. E

    Deriving the Reissner-Nordström metric

    I don't know how to do (a), so I decided to ignore it for now and just assume the result. Because ##j^a = 0## the Maxwell equations are ##\mathrm{d} \star F_{ab} = 0## and ##\mathrm{d} F_{ab} = 0##. For any two one forms, ##\frac{1}{2} \omega_a \wedge \eta_b = \omega_{[a} \eta_{b]}##, and so we...
  28. Arman777

    Tensor Calculations given two vectors and a Minkowski metric

    Let us suppose we are given two vectors ##A## and ##B##, their components ##A^{\nu}## and ##B^{\mu}##. We are also given a minkowski metric ##\eta_{\alpha \beta} = \text{diag}(-1,1,1,1)## In this case what are the a) ##A^{\nu}B^{\mu}## b) ##A^{\nu}B_{\mu}## c) ##A^{\nu}B_{\nu}## For part (a)...
  29. J

    I Ricci scalar for FRW metric with lapse function

    I need the Ricci scalar for the FRW metric with a general lapse function ##N##: $$ds^2=-N^2(t) dt^2+a^2(t)\Big[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta\ d\phi^2)\Big]$$ Could someone put this into Mathematica as I don't have it?
  30. mitochan

    I Coincidence of FLWR & CBR Homogeneity: Earth @ 0.0013c?

    The Earth is moving with respect to the CBR at a speed of 390 kilometers per second, I read in the article https://www.scientificamerican.com/article/how-fast-is-the-earth-mov/. Does FLWR metric coordinate space coincides with integrated local FRs where CBR is homogeneous, and the Earth is...
  31. Pouramat

    Einstein's Vacuum Exploring the Metric & Killing Vectors

    Einstein's vacuum solution metric: $$ ds^2 = -(1-\frac{2GM}{r})dt^2 +(1-\frac{2GM}{r})^{-1}dr^2+r^2 d\Omega^2 $$ which ##g_{\mu \nu}## can be read off easily. metric Killing vectors are: $$ K = \partial_t $$$$ R = \partial_\phi $$ How can I relate these to Maxwell equation?
  32. J

    A The "no metric, no nothing" view

    All normal relativists already adapted the point of view of "no metric, no thing"? Who are the relativists who don't? John Stachel who wrote the book Einstein from B to Z is a veteran 90 year old relativist. Einstein from 'B' to 'Z' - John Stachel - Google Books Which textbooks are...
  33. JD_PM

    Radial Geodesic in Kerr's metric

    The Kerr metric is given by \begin{align*} (ds)^2 &= -\left(1-\frac{2GMr}{\rho^2} \right)(dt)^2 - \frac{2GMar \sin^2 \theta}{\rho^2}(dt d\phi + d\phi dt) \\ &+ \frac{\rho^2}{\Delta}(dr)^2 + \rho^2 (d \theta)^2 + \frac{\sin^2 \theta}{\rho^2} \left[ \underbrace{(r^2+a^2)^2-a^2 \Delta \sin^2...
  34. JD_PM

    Getting a conserved charge out of the Kerr metric

    Compute the Komar integral for the Kerr metric \begin{equation*} J=-\frac{1}{8 \pi G} \int_{\partial \Sigma} d^2 x \sqrt{\gamma^{(2)}} n_{\mu} \sigma_{\nu} \nabla^{\mu} R^{\nu} \end{equation*} The Kerr metric is given by \begin{align*} (ds)^2 &= -\left(1-\frac{2GMr}{\rho^2} \right)(dt)^2...
  35. E

    I Are Derivatives of the Metric Different in Flat Spacetime?

    The general metric is a function of the coordinates in the spacetime, i.e. ##g = g(x^0, x^1,\dots,x^{n-1})##. That means that in the most general case we can't simplify an expression like ##\partial g_{\mu \nu} / \partial x^{\sigma}##. But, what about the special case of the flat spacetime...
  36. DuckAmuck

    I Einstein Field Eqns: East/West Coast Metrics

    My questions is: Depending on which metric you choose "east coast" or "west coast", do you have to also mind the sign on the cosmological constant in the Einstein field equations? R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} \pm \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} For example, if you...
  37. Bishal Banjara

    I Inverting the metric coefficients in the Schwarzschild line element

    Assuming the line element ##ds^s=e^{2\alpha}dt^2-e^{2\beta}dr^2-r^2{d\Omega}^2 ##as usual into the form ##ds^s=e^{-2\alpha}dt^2-e^{-2\beta}dr^2-r^2{d\Omega}^2##, I found that the ##G_{tt}## tensor component of first expression do not reconcile with the second one though, it fits for ##G_{rr}...
  38. snoopies622

    I Interpreting Schwarzschild Metric: Photon Falls Toward Black Hole

    As a photon falls radially toward the surface of a Schwarzschild black hole, dr/dt approaches zero. Does this mean that, from the viewpoint of a distant (Schwarzschild) observer, the photon slows down or that the distance covered by successive dr's is getting larger?
  39. steve1763

    A Find 2D Geometry of Line Element in Coordinates

    i'm trying to find what sort of 2-d geometry this system is in, I've been given the line element 𝑑𝑠2=−sin𝜃cos𝜃sin𝜙cos𝜙[𝑑𝜃2+𝑑𝜙2]+(sin2𝜃sin2𝜙+cos2𝜃cos2𝜙)𝑑𝜃𝑑𝜙 where 0≤𝜙<2𝜋 and 0≤𝜃<𝜋/2 Im just not sure where to start. I've tried converting the coordinates to cartesian to see if it yields a...
  40. T

    B Question on a property of a function which is not a metric

    We have the function d from VxV to another set(not necessarily R) for which the following properties are to be satisfied: i) d(x,y)=0<=>x=y ii)d(x,y)=d(y,x) iii)d(x,z)≤(d2(x,y)+d2(y,z))1/2 ∀ x,y,z ∈ V. What do you say? Would this function have interesting properties on a set and theorems to be...
  41. T

    I Exploring the Properties of Metric Spaces in Relation to Continuity

    Hello. The questions i make here in this thread are basically about like explanations of topics on metric spaces. We know about compactness, completeness, connectedness, separatedness, total boundedness of metric spaces. I know that continuity of the real line means that it has no gaps. What...
  42. LCSphysicist

    I What is the Minkowski metric tensor's trace?

    I am trying to follow the rule, that is, raising an index and the contract it. Be ##g_{\mu v}## the metric tensor in Minkowski space. Raising ##n^{v \mu}g_{\mu v}## and then, we need now to contract it. Now, in this step i smell a rat (i learned this pun today, hope this mean what i think this...
  43. Buzz Bloom

    I Q re de Sitter–Schwarzschild metric

    The equations below are from https://en.wikipedia.org/wiki/De_Sitter–Schwarzschild_metric#Metric . I am familiar with the dot on top of a variable as meaning d/dt, and the apostrophe as meaning d/dr (in this context). The dot on top of t, however, does not make any sense. I hope someone will...
  44. F

    A Demonstration of the Brans-Dicke's Lagrangian

    Helo, The Lagrangian in general relativity is written in the following form: \begin {aligned} \mathcal {L} & = \frac {1} {2} g ^ {\mu \nu} \nabla \mu \phi \nabla \nu \phi-V (\phi) \\ & = R + \dfrac {16 \pi G} {c ^ {4}} \mathcal {L} _ {\mathcal {M}} \end {aligned} with ## g ^ {\mu \nu}: ## the...
  45. J

    I Variation of Ricci scalar wrt derivative of metric

    I understand from the wiki entry on the Einstein-Hilbert action that: $$\frac{\delta R}{\delta g^{\mu\nu}}=R_{\mu\nu}$$ What is the following? $$\frac{\delta R}{\delta(\partial_\lambda g^{\mu\nu})}$$ Is there a place I could look up such GR expressions on the internet? Thanks
  46. P

    A Parallel plate capacitor in the Rindler metric

    Does anyone have a reference or solution for a parallel plate capacitor in the Rindler metric? I'm particularly interested in the case where the capacitor plates are in the xz or yz planes, z being the direction of the acceleration. The motivation is to get an idea how a transmission line...
  47. Pyter

    B Metric tensor for a uniformly accelerated observer

    Hello all, let's suppose we have, in a flat spacetime, two observers O and O', the latter speeding away from O, with an uniform acceleration ##a##. In the Minkowski spacetime chart of O, the world-line of O' can be drawn as a parable. We know that the Lorentz boost at every point of the...
  48. G

    A Metric Ansatz For Unifying All Forces In 11D?

    The ansatz for the 5D metric is \begin{equation} G_{\mu \nu}= g_{\mu \nu}+ \phi A_{\mu} A_{\nu}, \end{equation} \begin{equation} G_{5\nu} = \phi A_{\nu}, \end{equation} \begin{equation} G_{55} = \phi. \end{equation} This information was extremely enlightening for me, but what's the analogous...
  49. V

    I Riemannian Fisher-Rao metric and orthogonal parameter space

    Let ## \mathcal{S} ## be a family of probability distributions ## \mathcal{P} ## of random variable ## \beta ## which is smoothly parametrized by a finite number of real parameters, i.e., ## \mathcal{S}=\left\{\mathcal{P}_{\theta}=w(\beta;\theta);\theta \in \mathbb{R}^{n}...
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