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. E

    Diagonalization of metric matrix in general relativity

    1. Is its possible diagonalization of metric matrix (g_{uv}) in general relativity? 2. If we include imaginary numbers, can this help?
  2. jfy4

    Electron Falling in Kerr Metric: Release of 40% Rest Energy?

    I have here a quote from Hartle's Gravity, page 321: "The fraction of rest energy that can be released in making a transition from an unbound orbit far from an extremal black hole to the most bound innermost stable circular orbit is (1-1/\sqrt{3})\approx 42\%". My question is about...
  3. S

    Explore the Power of Concept Maps in Math: Metric Spaces to Geometry and Beyond

    I have posted this here for two reasons, one I think you all will really appreciate this as a teaching tool to hopefully increase your students understanding (and hopefully grades) & second because you all might find/make more concept maps on more advanced topics or with focus shifted in a...
  4. pellman

    Does the presence of torsion require a non-symmetric metric?

    If torsion = anti-symmetric part of the connection coefficients, and \Gamma_{\alpha\beta\gamma}=\frac{1}{2}(g_{\alpha\beta,\gamma}+g_{\alpha\gamma,\beta}-g_{\beta\gamma,\alpha}) then doesn't the metric have to have an antisymmetric component? The first two terms on RHS are together...
  5. M

    Riemannian Metric: Homework Ques on Compatibility w/Metric

    Homework Statement There's something about the Riemannian metric that I don't understand. I don't understand the compatibility with the metric. If X,Y,Z are vector fields, the Riemannian connection (I'll call it D) satisfies DX(g(Y,Z)) = g(DXY,Z) + g(Y, DXZ), where g is the Riemannian metric...
  6. michael879

    Contravariant metric components

    I realize this is a "simple" mathematical exercise, in theory, but I'm having a lot of trouble finding some algorithmic way to do it. The problem is this: I want to expand the contravariant metric tensor components g^{\mu\nu} in terms of the covariant metric tensor g_{\mu\nu}. The first order...
  7. T

    General Tensor contraction: Trace of Energy-Momentum Tensor (Einstein metric)

    Okay so I have: Eqn1) Tij=\rhouiuj-phij = \rhouiuj-p(gij-uiuj) Where Tij is the energy-momentum tensor, being approximated as a fluid with \rho as the energy density and p as the pressure in the medium. My problem: Eqn2) Trace(T) = Tii = gijTij = \rho-3p My attempt: Tr(T) = Tii...
  8. Z

    Question about time dilation in the schwarzchild metric?

    My first question is the following. Does the radial component of the schwarzchild metric account for just the radius of the body in study or is it the distance between the body and the observer, where the body is treated as a singularity (Point mass particle)? My second question is about how...
  9. B

    Definition of Normal (Intersection) Without Using a Metric

    Hi, Everyone: I am trying to understand the meaning of a statement that two embedded manifolds intersect normally*. The statement is made in a context in which any choice or existence of a metric is not made explicit, nor--from what I can tell-- implicitly either. If...
  10. B

    Understanding Disconnectedness in Countable Metric Spaces

    We know that every discrete metric space with at least 2 points is totally disconnected. Yet I read this: A MS that is countable with more than 2 pts is disconnected. Is it that I'm misreading this statement. It sounds like if it has 2 or less points it is connected? more means greater than.
  11. F

    Tortoise-like coordinate transform for interior metric

    Hello! When using the Schwarzschild exterior metric in the klein-gordon equation one can perform the standard tortoise(E-F) coordinate transform to yield a wave equation which has a well defined potential that is independent of the energy term. My understanding is that the motivation for this...
  12. J

    Tensor algebra with derivative of the metric

    I am trying to proove that the following relation: A_{\nu} \partial_{\mu} \partial^{\nu} A^{\mu} = A_{\nu} \partial^{\mu} \partial^{\nu} A_{\mu} The only way I found is by setting: A_{\nu} \partial_{\mu} \partial^{\nu} A^{\mu} = A_{\nu} \partial_{\mu} \partial^{\nu} g^{\mu \sigma}...
  13. B

    Topological and Metric Properties

    Can someone explain the difference between the two? 2 topo spaces are isometric if they have the same metric properties and homeometric if they have the same topological properties. If a space is homeo it is iso, but not vice verse. Which begs the conclusion that every topological...
  14. A

    Gravitational mass defect, weyl metric

    hello everyone, following the book of Landau&Lifsitz I managed to understand the Schwarzschild solution. At the end, it finds this formula for the mass of the spherical body generating the gravitational field: M=\frac{4\pi}{c^2} \int^a_0 \epsilon(r) r^2 dr in which \epsilon(r) is...
  15. Y

    How to obtain Kerr Metric via Spinors (N-P Formalism)

    How to obtain Kerr Metric via Spinors (Newman-Penrose Formalism)? I am a bit confused with Ray d'Inverno's Book. Why perform the coordinates transformation: 2r-1 -> r-1 + r*-1 I am bit confused of it. And I am a bit confused, too, of how to write out null tetrad...
  16. C

    Sketching curves on a plane with a given metric

    Homework Statement Consider a coordinate transformation from (t,x) to (u,v) given by t=u\sinh vx=u\cosh v Suppose (t,x) are coordinates in a 2-dimensional spacetime with metric ds^2=-dt^2+dx^2 Sketch, on the (t,x) plane, the curves u=constant and the curves v=constant.Homework Equations None...
  17. Z

    Having trouble writing down a metric in terms of metric tensor in matrix form?

    Can someone please explain to me how exactly you write down a metric, say the FLRW metric in matrix form. Say we have the given metric here. ds^2 = dt^2 - R(t)^2 * [dw^2 + s^2 * (dθ^2 + sin^2(θ)dΦ^2)] Thank you.
  18. P

    What does it mean by a Riemannian metric on a vector bundle?

    It's really a question about convention. Does such a metric have to be linear on each fiber?
  19. K

    Partial Derivative of Metric: Exploring Gravitational Action

    Could someone tell me what would be the partial derivative... \frac{ d g_{ab} }{ d g_{cd}} ?? Such expressions occur when one does variations of gravitational action... Note: For some reason, the d needed for derivative is not appearing in the post...although it was visible in the preview...
  20. F

    Physical implications of not-smooth metric derivative matching

    Hey all, My question pertains to interior metrics, for example the Schwarzschild interior metric given in post #5 of https://www.physicsforums.com/showthread.php?t=323684 The radial derivative of the first term, the dt^2 coefficient, matches the radial derivative of the Schwarzschild...
  21. B

    Proving T(x,y) is a Metric on Compact Set

    To show that some T(x,y) = something is a metric on a set for which it is compact, we have to prove that it respects the 3 axioms of distance. right?
  22. S

    Plotting Planets Orbits around Sun using Schwarzchild Metric

    Hello, I'm currently studying general relatively and am trying to plot orbits of planets around the sun using the schwarzchild metric. I've worked out the geodesic equations, working with c=1 to simplify things, and written a MATLAB script to plot trajectories, but I'm struggling to work out...
  23. Q

    Gravitons and the Field Theory Metric

    We know the background metric has the description; g_{\mu \nu}\sim\eta_{\mu \nu}+h_{\mu \nu} I would like to know what the physical meaning is of the difference then? h_{\mu \nu} - \eta_{\mu \nu} When I've read field theories describing gravitons, they are usually denoted as...
  24. TrickyDicky

    What is the physical meaning of metric compatibility and why is it important?

    What exactly is the physical meaning of the fact that the covariant derivative of the metric tensor vanishes?
  25. N

    Metric Tensor Questions: Understanding Hartle's "Gravity" Example 7.2

    Hi. This is example 7.2 from Hartle's "Gravity" if you happen to have it lying around. Metric of a sphere at the north pole The line element of a sphere (with radius a) is dS^{2}=a^{2}(d\theta^{2}+sin^{2}\theta d\phi ^{2}) (In (\theta , \phi ) coordinates). At the north pole \theta = 0 and at...
  26. V

    Derivative of metric tesor and its trace

    I would like to ask, how these identities are true \partial_{\mu}(-g)=(-g)g^{\alpha\beta}\partial_{\mu}g_{\alpha\beta} and \partial_{\mu}g^{\alpha\beta}=-g^{\alpha\lambda}g^{\beta\rho}\partial_{\mu}g_{\lambda\rho} Sorry I meant" derivative of metric tensor and its determinant", I was able to...
  27. L

    Kerr Metric Confusion: Problem 1 on Page 138

    Hi. I'm trying problem 1 on p138 of this http://arxiv.org/PS_cache/gr-qc/pdf/9707/9707012v1.pdf Now when I try and get the Euler Lagrange equation for \phi I get (the Kerr metric in BL coordinates can be found at the bottom of p77) \frac{\partial L}{\partial \phi} = \frac{d}{d \tau}...
  28. M

    Continuity of Metric Spaces: Does the Distance Between Points Remain Consistent?

    Homework Statement Let (X,d) be a metric space and let {x_n} be a sequence in X converging to a. Show that d(b, x_n ) ->d(b,a) Homework Equations The Attempt at a Solution For every eps > 0 there is an N such that d(x_n,a) < eps for all n>= N But where do I go from here...
  29. Y

    How to calculate the contraction of metric tensor g^ab g_ab

    I wish I could calculate the contraction: gabgab I wish someone could show me how to get n! Unfortunately, I find it difficult, for I am not familiar with Tensor Algebra ... My wrong way to calculate it: gabgab= gabgba (since gab is symmetric) = δaa = 1Why is it wrong?
  30. B

    Is d((xn), (yn)) = lim d(xn, yn) a metric for Cauchy sequences in (X, d)?

    If (xn) and (yn) are two Cauchy sequences in a metric space (X, d), and we define d((xn), (yn)) = lim d(xn, yn). Is this a metric on the set of all Cauchy sequences? I'm thinking yes since all 3 properties work.
  31. C

    Metric Spaces, Triangle Inequality

    I have the following question on metric spaces Let (X,d) be a metric space and x1,x2,...,xn ∈ X. Show that d(x1, xn) ≤ d(x1, x2) + · · · + d(xn−1, xn2 ), and d(x1, x3) ≥ |d(x1, x2) − d(x2, x3)|. So the first part is simply a statement of the triangle inequality. However, the metric...
  32. Demon117

    Triangle Inequality for d(m,n) Metric Proof

    1. For integers m and n, let d(m,n)=0 if m=n and d(m,n) = 1/5^k otherwise, where k is the highest power of 5 that divides m-n. Show that d is indeed a metric. 2. The attempt at a solution Here is what I have come up with: PROOF: Clearly by definition d(m,n) = 0 iff m=n and d(m,n)>0...
  33. T

    Understanding the Usual Metric on R - {0}: A Question from a Homework Statement

    Homework Statement I was working on a problem and think I might have run across an issue. Is the usual metric defined on R - {0}? (Where R is the real numbers) Reworded, can I say that I have a space R - {0} with the usual metric on it? Thank you.
  34. pellman

    Does the metric have to be symmetric? Why?

    Why must we have g_{\mu\nu}=g_{\nu\mu}? What are the physical consequences if this did not hold?
  35. Demon117

    Proof about compact metric spaces.

    1. Let M be a compact metric space. If r>0, show that there is a finite set S in M such that every point of M is within r units of some point in S. 2. Relevant theorems & Definitions: -Every compact set is closed and bounded. -A subset S of a metric space M is sequentially compact...
  36. L

    Reissner- Nordstrom metric- weak energy condition

    Hey, For a problem I am working on I need to know if the energy-momentum tensor of the Reissner- Nordstrom metric obeys the weak energy condition. Since I need the result for the following calculations, I just want to be sure that it is correct. Does anyone of you know the correct answer and...
  37. C

    How to make a hodge dual with no metric, only volume form

    Hey guys! I am going crazy... most books don't cover this and instead assume that the manifold is Riemannian or pseudo-Riemannian and has a metric tensor defined on it. I want a "generalized" hodge star. I have an orientable smooth manifold, that's IT. I have heard that there is a way to...
  38. W

    Please help. Imposing a metric to preserve distance

    If we use the mapping (r,phi)---->(x,y)= (2tan(r/2)cos(phi), 2tan(r/2)sin(phi)) Which metric do we have to impose on R^2 in order that the mapping preserves distance. Any help would be greatly appreciated. Thank you very much
  39. B

    Compactness in Metric Spaces: Is It Possible?

    I've never actually seen a proof that a space is compact just from the definition. In metric spaces it was usually after the notion of closed and bounded or sequential compactness was introduced. For example is there a way to prove [a,b] is compact (with the usual topology on the real numbers)...
  40. G

    Reissner-Nordström metric for magnetic fields

    hey, does anyone of you know the reissner-nordström metric, if there is a magnetic field instead of the coulomb field of a charged gravitating body? i just need the formula, but i could not find it in the internet, maybe someone here knows it? sorry for my english, but i am german :uhh:
  41. T

    Define Metric for Set of Reals in n Dimensions

    How does one redefine the metric of a given set, such as the reals? I thought it would be an interesting concept to have a metric defined like so: d_X:X^n \times X^n \to \Re (x_1, x_2, x_3, \cdots, x_n), (y_1, y_2, y_3, \cdots, y_n) \mapsto \sum^{n}_{i=1}{\frac{x_i+y_i}{2}} Does it have to be...
  42. J

    How Do You Convert Cubic Centimeters to Cubic Meters?

    How many cubic meters are in 5.43 x 10^6 cm^3? There are .001 meters in the centimeter. m= meter cm= centimeter This is what I did: 5.43 x 10^6 cm^3 (.01m/1 cm)^3 And then did the calculations, which gave me: .0000043 x 10^6 m^3 or 4.3 m^3. Not sure if I set up the...
  43. I

    Dark matter, dark energy, and the Kerr metric?

    I’m sorry, but I find dark matter and dark energy problematic. It’s hard to think of a Universe made up of about 95 % of stuff we have no idea about, except that maybe dark matter and dark energy have some properties. So I’m thinking maybe there’s something wrong with the data, but I can’t...
  44. S

    FRW metric for closed universe

    Hi, I'm new to Physics Forum and wasn't really sure where to post this since its not strictly speaking a homwork question. So if it happens to be in the wrong place I apologise. I was looking through some lecture notes from when I did my Physics degree years ago and come across a problem...
  45. M

    I proving l^2 is a complete metric space

    Homework Statement Prove that the sequence space l^2 (the set of all square-summable sequences) is complete in the usual l^2 distance. Homework Equations No equations.. just the definition of completeness and l^2. The Attempt at a Solution I have a sample proof from class to show...
  46. TrickyDicky

    Exploring FRW Metric Symmetries in Spacetime

    What are the symmetries determined by FRW spacetime? I guess they include Lorentz symmetry, rotationally and translationally symmetries, but not time symmetry. Is this right? Thanks
  47. J

    Why do some solutions in general relativity have no mass term?

    I'm not very fund of the subject, but what i know is that the Schwarzschild metric and other known solutions for Einstein's action have some constant representing mass. However, I encountered some solutions where no such constant existed. Can someone explain to me what does this mean exactly? In...
  48. E

    An Easy Metric for Einstein Field Equations

    So I am an engineering graduate trying to teach myself some general relativity. I have tried to solve the Einstein Field equations for a wormhole metric and some others. After pages and pages of calculating Christoffel Symbols, Riemann Tensors, Ricci Tensors and Scalars, and so on, I end...
  49. M

    Transformation on the minkowsky metric

    hi, in my textbook there ist a problem, i cannot solve: let T be a linear bijective map from R^4 to R^4, which preserves the light cone. show: T* ds^2 = (constant)^2 * ds^2, where ds^2 ist the minkowsky metric and T* ist the pullback of the metric. can someone show how to do it. michiherlin
  50. Z

    What are physics constant in Kerr metric?

    1. What are the value of physics constant in Kerr metric, including G, M, c, a, r, or others? I expect to simplify Gamma 2. why g_compts[1,4] has element and not [4,1]? 3. Some book assume G = c = 1, what is the meaning of this setting? 4. Different material have different metric, are...
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