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. Math Amateur

    MHB Metric Spaces & Compactness - Apostol Theorem 4.28

    I need help with the proof of Theorem 4.28 in Tom Apostol's book: Mathematical Analysis (2nd Edition). Theorem 4.28 reads as follows:In the proof of the above theorem, Apostol writes: " ... ... Let m = \text{ inf } f(X). Then m is adherent to f(X) ... ... " Can someone please explain to me...
  2. binbagsss

    Levi-Civita Connection & Riemannian Geometry for GR

    Conventional GR is based on the Levi-Civita connection. From the fundamental theorem of Riemann geometry - that the metric tensor is covariantly constant, subject to the metric being symmetric, non-degenerate, and differential, and the connection associated is unique and torsion-free - the...
  3. J

    Spacetime Interval & Metric: Equivalent?

    This may seem an odd question but it will clear something up for me. Are "The spacetime interval is invariant." and the "The spacetime metric is a tensor." exactly equivalent statements? Does one imply more or less information than the other? Thanks!
  4. K

    How Do You Convert Gallons to Metric Units and Calculate Paint Thickness?

    There are .67 gallons of paint in a can. A. How many cubic meters of paint are in the can? B. How many liters of paint are in the can? C. Imagine that all of this paint is used to apply a coat of uniform thickness to a wall of area 13m^2. What is the thickness of the layer of wet paint in metric...
  5. G

    Calculate metric tensor in terms of Mass

    Homework Statement Suppose everything is moving slowly, How can we find the metric tensor in GR in terms of the mass contained. Homework Equations I understand in case of everything moving slowly only below equation is relevant - R00 - ½g00R = 8πGT00 = 8πGmc2 The Attempt at a Solution None.
  6. C

    Is a Pseudo-Riemann Metric Intrinsic to General Relativity?

    In considering special relativity as a limiting case of the general theory (without matter or curvature) the question arose as to whether the pseudo-riemann nature of the SR metric is actually an independant (essentially experimentally determined) assumption/property or derivable from the...
  7. T

    Understanding Einstein Field Equation & Metric Tensor

    Hi guys. I am trying to understand einstein field equation and thus have started on learning tensor. For metric tensor, is it just comprised of two contra/covariant vectors tensor product among each other alone or does it requires an additional kronecker delta? I am confused about the idea...
  8. E

    What Does a Rotating Mass in Kerr Metric Rotate With Respect To?

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  9. N

    Experimental determination of the metric tensor

    Does anyone know a reference with a discussion on the experimental determination of the metric tensor of spacetime? I only know the one in "The theory of relativity" by Møller, pages 237-240. https://archive.org/details/theoryofrelativi029229mbp
  10. M

    Understanding the Metric Tensor: A 4-Vector Perspective

    Some subtleties of the metric tensor are just becoming clear to me now. If I take ##g_{\mu\nu}=diag(+1,-1,-1,-1)## and want to write ##\partial_\mu\phi^\mu##, it would be ##\partial_0\phi^0 -\partial_i\phi^i##, correct? ##\phi## is a 4-vector.
  11. binbagsss

    GR: FRW Metric relation between the scale factor & curvature

    Mod note: OP warned about not using the homework template. I have read that 'a(t) determines the value of the constant spatial curvature'.. Where a(t) is the scale factor, and we must have constant spatial curvature - this can be deduced from the isotropic at every point assumption. I'm trying...
  12. binbagsss

    Simple indice question -- metric, traces

    Mod note: OP warned about not using the homework template. Let ##g_{ac}## be a 3-d metric. So the trace of a metric is equal to its dimension so I get ##g_{ac}g^{ac}=3## But I'm a tad confused with the expression : ##g^{ac}g_{ad}##=##delta^{c}_{d}## I thought it would be ##3delta^{c}_{d}###...
  13. binbagsss

    Form of Rienmann Tensor isotrpic & homogenous metric quick Question

    Context: Deriving the maximally symmetric- isotropic and homogenous- spatial metric I've seen a fair few sources state that the Rienamm tensor associated with the metric should take the form: * ##R_{abcd}=K(g_{ac}g_{bd}-g_{ad}g_{bc})## The arguing being that a maximally symmetric space has...
  14. P

    Ricci tensor of schwarzschild metric

    In schwarzschild metric: $$ds^2 = e^{v}dt^2 - e^{u}dr^2 - r^2(d\theta^2 +sin^2\theta d\phi^2)$$ where v and u are functions of r only when we calculate the Ricci tensor $R_{\mu\nu}$ the non vanishing ones will only be $$R_{tt}$$,$$R_{rr}$$, $$R_{\theta\theta}$$,$$R_{\phi\phi}$$ But when u and v...
  15. Berlin

    Does the Square Root of the Inverse Metric Unify Geometry and Physics?

    Anyone noticed this paper: Square Root of Inverse Metric: The Geometry Background of Unified Theory? Authors: De-Sheng Li, arXiv:1412.2578 ? The author tries to construct the square root of the inverse metric, based on a product of a fermion field and a framefield. Somehow the Standard model...
  16. C

    Null geodesics of the FRW metric

    When working with light-propagation in the FRW metric $$ds^2 = - dt^2 + a^2 ( d\chi^2 + S_k(\chi) d\Omega^2)$$ most texts just set $$ds^2 = 0$$ and obtain the equation $$\frac{d\chi}{dt} = - \frac{1}{a}$$ for a light-ray moving from the emitter to the observer. Question1: Do we not strictly...
  17. Einj

    Understanding the FRW Metric: Exploring Physical and Comoving Coordinates

    Hello everyone, I already know that the solution to this question is obvious but I can't find it. Consider an expanding universe following the FRW metric ds^2=-dt^2-a^2(t)dx^2 (1 space dimension for simplicity). We know that the physical spatial distance x_p is related to the comoving spatial...
  18. C

    Determining Metric of Space-Time (H.P. Robertson 1949)

    In a paper published in Reviews of Modern Physics in 1949, http://journals.aps.org/rmp/pdf/10.1103/RevModPhys.21.378 , H.P. Robertson provided an analysis of the physical implications of the Michelson/Morley, Kennedy and Thorndike, and Ives and Stilwell experiments which seems definitive with...
  19. S

    How did Einstein derive the meaning of Schwarzchild's metric

    I have been recently working with Schwarzschild's solution: ds2= - (1- (2GM/rc2))c2dt2 + dr2/(1-2GM/rc2) + r2(dθ2 + sin2(θ)d∅2) Now, when deriving the various general relativistic tensors for this metric such as the Ricci tensor, I found the calculations to be painfully tedious and monstrous...
  20. Breo

    Doubt: Why Quadratic in Matrix but Power 4 in Einstein-Rosen Metric?

    I have a doubt since I see the next equation and the corresponding matrix: $$ ds^2 = \Bigg( \frac{1-\frac{r_s}{4\rho}}{1+\frac{r_s}{4\rho}}\Bigg)^2 dt^2 - \Big(1+\frac{r_s}{4\rho}\Big)^4 (d\rho^2 + p^2 d\Omega_2^2) $$$$ g_{\mu\nu} = \left( \begin{array}{ccc} \Bigg(...
  21. Breo

    What is the Unique Metric Tensor in this Line Element?

    Hello, this is the metric I am talking about: $$ ds^2= (dt - A_idx^i)^2 - a^2(t)\delta_{ij}dx^idx^j $$ I never see one like this. How the metric tensor matrix would be?
  22. P

    Israel Wilson Perjes Metric: Tetrad Formalism Reference

    Is there any book or reference perhaps on string theory or superstring theory or even advanced general relativity that treats the Israel Wilson Perjes metric using the tetrad formalism in details, i.e, 1-forms and so? (Not spinors methos) I have ran across many papers that just place the spin...
  23. B

    Can g_00 of the metric tensor depend on time

    In SRT, the line element is ##c^2ds^2 = c^2dt^2 - dx^2 -dy^2-dz^2## and ##g_{00} = 1## (or ##-1## depending on sign conventions). In the Schwarzschild metric we have g_{00}=(c^2-\frac{2 GM}{r}) . So in the first example, ##g_{00}## is constant, in the second it depends on another coordinate...
  24. M

    MHB Show that it is metric and the measurable is 0

    Hey! :o In a space of finite measure, if $f$ and $g$ are measurable we set $\rho (f,g)=\int \frac{|f-g|}{1+|f-g|}d \mu$. Show that $\rho$ is metric and that $f_n \rightarrow f$ as for $\rho$ if and only if $\forall c>0$ we have that $\mu(\{|f_n-f|>c\})\rightarrow 0$.What does "$f_n \rightarrow...
  25. Orion1

    Non-rotational and rotational metric tensors

    General Relativity... Non-rotational spherically symmetric body of isotropic perfect fluid Einstein tensor metric element functions: g_{\mu \nu} = \left( \begin{array}{ccccc} \; & dt & dr & d \theta & d\phi \\ dt & g_{tt} & 0 & 0 & 0 \\ dr & 0 & g_{rr} & 0 & 0 \\ d\theta & 0 & 0 & g_{\theta...
  26. K

    Motion along a curved path -- introductory

    1. "William Tell is said to have shot an apple off his son's head with an arrow. If the arrow was shot with an initial speed of 55m/s and the boy was 15m away, at what launch angle did Bill aim the Arrow? (Assume that the arrow and the apple are initially at the same height above the...
  27. N

    Partial derivative with respect to metric tensor

    \mathcal{L}_M(g_{kn}) = -\frac{1}{4\mu{0}}g_{kj} g_{nl} F^{kn} F^{jl} \\ \frac{\partial{\mathcal{L}_M}}{\partial{g_{kn}}}=-\frac{1}{4\mu_0}F^{pq}F^{jl} \frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})=+\frac{1}{4\mu_0} F^{pq} F^{lj} 2 \delta^k_p \delta^n_j g_{ql} I need to know how...
  28. arpon

    How to draw a 2D space in 3D Euclidean space by metric tensor

    Suppose, I know the metric tensor of a 2D space. for example, the metric tensor of a sphere of radius R, gij = ##\begin{pmatrix} R^2 & 0 \\ 0 & R^2\cdot sin^2\theta \end{pmatrix}## ,and I just know the metric tensor, but don't know that it is of a sphere. Now I want to draw a 2D space(surface)...
  29. R

    How does one get time dilation, length contraction, and E=mc^2 from spacetime metric?

    How does one get time dilation, length contraction, and E=mc^2 from the spacetime metric? Suppose all that you are given is x12 + x22 + x32 - c2t2 = s2 How do you derive time dilation, length contraction, and E=mc^2 from this? What is the most direct way to do this?
  30. RCopernicus

    Understanding Minkowski Space Metrics: The Sign Reversal Mystery Explained

    I've never seen a satisfactory explanation of the metrics used in a calculation of distance in Minkowski space. In Euclidean space, the distance is: ds^2 = dx^2 + dy^2 + dz^2 But in Minkowski space, the distance is ds^2 = (dt * c)^2 - dx^2 - dy^2 - dz^2 Why are the signs reversed? This implies...
  31. stevendaryl

    Growing Black Hole Metric Approximation: 2MG/c^2

    This is a question inspired by the "Golf Ball" thread, which is no longer open for comments, I guess. For a black hole of constant mass, the metric external to the black hole can be written in Schwarzschild metric, which is characterized by the constant M, and the corresponding radius 2 M...
  32. D

    Are there any metric spaces with no Cauchy sequences?

    A metric space is considered complete if all Cauchy sequences converge within the metric space. I was just curious if you could have a case of a metric space that doesn't have any Cauchy sequences in it. Wouldn't it be complete by default? When trying to think of a space with no cauchy...
  33. P

    Deriving Metric $$g_{ij}$$ w/ Respect to Time t

    Hello, I have a simple question about deriving $$g_{ij} \frac{\partial x^i}{\partial t}\frac{\partial x^j}{\partial t}$$ with respect to time t. I have noticed that the first term after derivation turns out to be$$ \frac{\partial g_{ij}}{\partial x^k} \frac{\partial x^k}{\partial...
  34. S

    Questions about Traversable Wormhole Metric

    First of all, the metric I am referring to is this one: ds2= -c2dt2 + dl2 + (k2 + l2)(dᶿ2 + sin2(ᶿ)dø2) where k is the radius of the throat of the wormhole. (sorry for the small Greek letters) Now I have two questions about this solution to Einstein's equations: 1. What does the coordinate l...
  35. R

    Euclidean metric and non-Cartesian systems

    OK. A problem is simple and probably very stupid... In many (most of) maths books things like Euclidean metric, norm, scalar product etc. are defined in Cartesian coordinate system (that is one which is orthonormal). But what would happen if one were to define all these quantities in...
  36. K

    Metric in Manifold Homework: 3-sphere in 4D Euclidean Space

    Homework Statement I am confused if there is any standard way to check what should be the line element $$ds^{2}$$ when the dimensions are more than three ( since we don't have the option to draw things as we usually do in case of 3d or less dimensional cases. I am following Hobson's book. I am...
  37. Ravi Mohan

    Event-horizon from the blackhole metric

    What is the most general method of obtaining the event-horizon from the given black hole metric. Let us consider Kerr black hole in Kerr coordinates given by ds^2 = -\frac{\Delta-a^2sin^2\theta}{\Sigma}dv^2+2dvdr -\frac{2asin^2\theta(r^2+a^2-\Delta)}{\Sigma}dvd\chi-2asin^2\theta d\chi dr +...
  38. S

    Temporal components in metric tensors

    As you may know, the metric tensor for 3D spherical coordinates is as follows: g11= 1 g22= r2 g33= r2sin2(θ) Now, the Minkowski metric tensor for spherical coordinates is this: g00= -1 g11= 1 g22= r2 g33= r2sin2(θ) In both of these metric tensors, all other elements are 0. Now...
  39. B

    Lorentz transformations and Minkowski metric

    I am attempting to read my first book in QFT, and got stuck. A Lorentz transformation that preserves the Minkowski metric \eta_{\mu \nu} is given by x^{\mu} \rightarrow {x'}^{\mu} = {\Lambda}^\mu_\nu x^\nu . This means \eta_{\mu \nu} x^\mu x^\nu = \eta_{\mu \nu}x'^\mu x'^\nu for all x...
  40. X

    Homemorphism of two metric space

    If I create a bijective map between the open balls of two metric spaces, does that automatically imply that this map is a homemorphism?
  41. S

    What is the metric tensor of the 4-sphere?

    After my recent studies of the curvature of the 2- sphere, I would like to move on to Minkowski space. However, I can not seem to find the metric tensor of the 4 sphere on line, nor can I seem to think of the vector of transformation properties that I would use to derive the metric tensor of the...
  42. ChrisVer

    Metric Tensor Components: Inverse & Derivatives

    I have one question, which I don't know if I should post here again, but I found it in GR... When you have a metric tensor with components: g_{\mu \nu} = \eta _{\mu \nu} + h_{\mu \nu}, ~~ |h|<<1 (weak field approximation). Then the inverse is: g^{\mu \nu} = \eta^{\mu \nu} - h^{\mu \nu}...
  43. B

    GR: Metric, Inverse Metric, Affine Connection Caluculation Help

    Homework Statement Consider the Schwarschield Metric in four dimensional spacetime (M is a constant): ds2 = -(1-(2M/r))dt2 + dr2/(1-(2M/r)) + r2(dθ2 + sin2(θ)dø2) a.) Write down the non zero components of the metric tensor, and find the inverse metric tensor. b.) find all the...
  44. S

    Is curvature possible for a 2D metric?

    I was recently trying to test something out with the Riemann tensor. I used only 2 dimensions for simplicity sake. As I was deriving the Riemann tensor, I noticed that it looked as if all of the elements were going to come out to be 0 (which they all did). Therefore, this coordinate system is...
  45. ChrisVer

    Schwarz's Metric Gamma: Inertial Coordinates & Time Dilation

    Can the above logic be applied to Schw. Metric as well? Suppose I have an object moving with a radial velocity v=const, then can I do the same to derive the Schwarchild time dilation as in the Minkowski? dr = v ~ dt ds^{2} = [K - \frac{v^2}{K} ] dt^2 So \gamma ^{-1} = \sqrt{K} [1 -...
  46. shounakbhatta

    The question: What are the components of the gab metric in Kaluza-Klein theory?

    Hello, I am trying to understand Kaluza Klein theory on the five dimensional unification. It was mentioned over there: " Of the 15 components of gαβ, five had to get a new physical interpretation, i.e. gα5 and g55; the components gik, i,k = 1,...,4, were to describe the gravitational field...
  47. W

    What is a metric for uniformly moving frame?

    The Schwarzschild Metric has a form: ##ds^2 = Kdt^2 - 1/K dr^2 - r^2dO^2## where: K = 1 - a/r; There is a time scaled by K, but a space radially by 1/K. This is a typical time dilation and a space contraction, which is known from SR, but the Schwarzschild metrics is spherically...
  48. J

    Local Conformal Transformations:Coordinate or metric transformations?

    Hello, I'm wondering what the exact definition of a local conformal transformation is, in the context of General Relativity (/Shape Dynamics) To be more precise: 1. Are local conformal transformations coordinate transformations or scalar transformations of the metric? 2. If they are...
  49. M

    Confusion with Dot Product in Polar Coordinates with the Metric Tensor

    Alright, so I was reading up on tensors and such with non-Cartesian coordinate systems all day but now I'm a bit tired an confused so you'll have to forgive me if it's a stupid question. So to express the dot product in some coordinate system, it's: g(\vec{A}\,,\vec{B})=A^aB^bg_{ab} And, if...
  50. ChrisVer

    Mass in Schwarzschild's Metric: Exploring Vacuum Solutions

    I think this will be a quick question... If the Schw's metric is a solution of the vacuum, then what does the mass M_0 in the metric correspond to? I thought it was the mass of the star... but if that's true then why is it a vacuum solution? Or is it vacuum because it describes the regions...
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