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.
We all know that The gradient of a scalar-valued function ##f(x)## in ##IR^n## is a vector field ##V_\mu(x)=\partial_\mu f(x)##, Such a vector field is said to be conservative. Not all vector fields are conservative. A conservative vector field should meet certain constraints...
In this popular science article [1], they say that if our universe resulted to be non-uniform (that is highly anisotropic and inhomogeneous) then the fundamental laws of physics could change from place to place in the entire universe. And according to this paper [2] anisotropy in spacetime could...
I have the following question to solve:
Use the metric:
$$ds^2 = -dt^2 +dx^2 +2a^2(t)dxdy + dy^2 +dz^2$$
Test bodies are arranged in a circle on the metric at rest at $$t=0$$.
The circle define as $$x^2 +y^2 \leq R^2$$
The bodies start to move on geodesic when we have $$a(0)=0$$
a. we have...
Are there non-smooth metrics for spacetime (that don't involve singularities)?
I found this statement in a discussion about the application of local Lorentz symmetry in spacetime metrics:
Lorentz invariance holds locally in GR, but you're right that it no longer applies globally when gravity...
I started by expanding ##dx## and ##dt## using chain rule:
$$dt = \frac{dt}{dX}dX+\frac{dt}{dT}dT$$
$$dx = \frac{dx}{dX}dX+\frac{dx}{dT}dT$$
and then expressing ##ds^2## as such:
$$ds^2 =...
I was discussing this paper with a couple of physicists colleagues of mine (https://arxiv.org/abs/2011.12970)
In the paper, the authors describe "spacetimes without symmetries". When I mentioned that, one of my friends said that no spacetime predicted or included in the theory of relativity...
Hi, I am reading through my lecture notes - I haven't formally covered killing vectors but it was introduced briefly in lectures.
Reading through the notes has highlighted something I am not sure about when it comes to co-ordinate transformations.
Q1.Can someone explain how to go from...
Dear PF Forum,
It's been a while since I logged in here. And I really do appreciate all the answers that I've been getting here.
Now, I wonder. Is there any standardization for 1 light year distance?
Is it 10 trillion kilometers, or
299,792,458 * 60 * 60 * 24 * 365.256 = ...
Consider a hypothetical five dimensional flat spacetime ##\mathbb{R}^5## with coordinates ##x, y, z, w, t##.
Now imagine that the hypersurface ##\Sigma =\mathbb{R}^3## of ##x, y, z## moves with constant rate ##r## along the coordinate ##w##, i.e. ##dw/dt=r##. Assuming that ##t \in (-\infty, +...
In Dirac's "General Theory of Relativity", he begins Chap 16, with "Let us consider a static gravitational field and refer it to a static coordinate system. The ##g_{\mu\nu}## are then constant in time, ##g_{\mu\nu,0}=0##. Further, we must have ##g_{m0} = 0, (m=1,2,3)##."
It's obvious that...
Once having converted the FLRW metric from comoving coordinates ##ds^2=-dt^2+a^2(t)(dr^2+r^2d\phi^2)## to "conformal" coordinates ##ds^2=a^2(n)(-dn^2+dr^2+r^2d\phi^2)##, is there a way to facilitate solving for general geodesics that would otherwise be difficult, such as cases with motion in...
"Spherical symmetry requires that the line element does not vary when##\theta## and##\phi## are varied,so that ##\theta##and ##\phi##only occur in the line element in the form(##d\theta^2+\sin^{2}\theta d\phi^2)##"
I wonder why:
"the line element does not vary when##\theta## and##\phi## are...
From Wikipedia article about Hyperbolic motion, I have the following coordinate equations of motion for Bob in his accelerated frame:
##t(T)=\frac{c}{g} \cdot \ln{(\sqrt{1+(\frac{g \cdot T}{c})^2}+\frac{g \cdot T}{c})} \quad (1)##
##x(T)=\frac{c^2}{g} \cdot (\sqrt{1+(\frac{g \cdot T}{c})^2}-1)...
In describing the spacetime around a massive, spherical object, one would use the Schwarzschild Metric. What metric would instead be used to describe the spacetime around multiple massive bodies? Say, for example, you want to calculate the Gravitational Time Dilation experienced by a rocket ship...
I've been trying to find a way to calculate Gaussian curvature from a 4D metric tensor. I found a program that does this in Mathematica using the Brioschi formula. However, this only seems to work for a 2D metric or formula (I would need to use something with more dimensions). I've found...
I'm wondering if there is a way to find the proper volume of the warped region of the Alcubierre spacetime for a constant ##t## hypersurface. I can do a coordinate transformation ##t=τ+G(x)##, where ##G(x)=\int \frac{-vf}{1-v^2f^2}dx##. This eliminates the diagonal and makes it so that the...
Q. Calculate the linearised metric of a spherically symmetric body ##\epsilon M## at the origin. The energy momentum tensor is ##T_{ab} = \epsilon M \delta(\mathbf{r}) u_a u_b##. In the harmonic (de Donder) gauge ##\square \bar{h}_{ab} = -16\pi G \epsilon^{-1} T_{ab}## (proved in previous...
The Robertson-Walker-Metric is given by
To calculate the Friedmann equations ist is choosed
with
despite Minkowskis, Schwarzschilds and Kerrs
What is the reason for this difference?
Tanu
The FLRW metric has been introduced to characterize the homogeneity and isotropy of the Universe and accordingly to obtain "easy" manageable solutions in Friedmann equations.
The FLWR metric is
where the LHS can be written as where is the proper time (despite we know that time is...
I'm interested in describing a 6-dimensional space of which three are compactified to small circles. Globally this space looks 3-dimensional, like a 2-dimensional cylinder looks 1-dimensional globally.
Kaluza and Klein did a similar thing in the context of 4-dimensional spacetime. They extended...
In the absence of a metric, we can not raise and lower indices at will.
There are two sorts of Christoffel symbols, Christoffels of the first kind, ##\Gamma^a{}_{bc}## in component notation, and Christoffel symbols of the second kind, ##\Gamma_{abc}##. What's the relationship between the two...
5/18/22
I am an MS in physics.
I need to find out if the following CONFORMAL
METRIC produces zero or nonzero curvature?
I suspect the curvature is zero, but others
have said it's probably not? MAXIMA
sometimes says it is, and other times produces
a Ricci scalar that looks like the FRW scalar...
The Schwarzschild metric implies a potential different from that of Newtonian gravity. Is there a relationship between it and the process by which particles can be absorbed by other particles?
(I haven't studied QFT yet)
I'm working through some things with general relativity, and am trying to solve for my equations of motion from the Schwarzschild Metric. I'm new to nonlinear pde, so am not really sure what things to try. I have 2 out of my 3 equations, for t and r (theta taken to be constant). At first glance...
For the dimensions of a right cylinder, I am given three significant digits for the diameter (17.4 mm) and the height (50.3 mm). The formula for the volume of a right cylinder is V = Pi x r^2 x h, which would lead here to Pi x (17.4 mm / 2)^2 x 50.3 mm = 11,960.69354 mm^3 before rounding to 3...
Homework Statement:: Please see below.
Relevant Equations:: Please see below.
I am trying to find a reference to a textbook or a paper that details the following time-invariance Kaluza-Klein metric:
\begin{equation}...
##d'## is a metric on ##X## because it satisfies the axioms of metrics:
Identity of indiscernibles:
##x=y\Longleftrightarrow d(x,y)=0\Longleftrightarrow \sqrt{d(x,y)}=\sqrt{0}##
Symmetry: ##d(x,y)=d(y,x)\Longrightarrow \sqrt{d(x,y)}=\sqrt{d(y,x)}##
Triangle inequality: ##d(x,z)\leq...
So, there are a fair amount of metrics designed with a zero value for the cosmological constant in mind. I was wondering if there was some method to modify metrics to account for a nonzero cosmological constant. Say, for instance, the Schwarzschild metric due to its relative simplicity. A...
M. Blennow's book has problem 2.18:
Show that the contracted Christoffel symbols ##\Gamma_{ab}^b## can be written in terms of a partial derivative of the logarithm of the square root of the metric tensor $$\Gamma_{ab}^b=\partial_a\ln{\sqrt g}$$I think that means square root of the determinant of...
I know some basic GR and encountered the Schwarzschild metric as well as the Riemann tensor. It is known that for maximally symmetric spaces there is a corresponding Riemann tensor and thus Ricci scalar.
Question. How do you calculate the Ricci scalar ##R## and cosmological constant ##\Lambda##...
I'm still confused about the notation used for operations involving tensors.
Consider the following simple example:
$$\eta^{\mu \sigma} A_{\mu \nu} = A_{\mu \nu} \eta^{\mu \sigma}$$
Using the rules for raising an index through the (inverse) metric tensor ##\eta^{\mu \sigma}## we get...
I am having trouble calculating the extrinsic curvature (12) in the following paper: https://arxiv.org/pdf/gr-qc/0310107.pdf
Specifically, I am unsure of what term to plug in for the induced metric h_{ab} in (8). If I am calculating the \sigma term in (12) is h_{ab} all of (4)?
Also I would like...
I can't figure out how to transform the coordinates to get to the given metric \begin{align*}ds^2 = \cos x (dy^2 - dx^2) + 2\sin x dx dy \end{align*} for a 2-torus. Initially I parameterised it by two angles ##\theta## (around the ##z## axis) and ##\phi## (around the torus axis), to write...
In the Reissner–Nordström metric, the charge ##Q## of the central body enters only as its square ##Q^2##. The same is true for the Kerr-Schild form. This would seem to imply that all effects are even functions of ##Q##. For example, the gravitational time dilation is often written as
$$\gamma =...
if a metric like ##ds^2=dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2 ## is given, we know it corresponds to a sphere in spherical coordinates .
if you are given an arbitrary metric with two variables for example ##ds^2=\frac{du^2}{u}+dv^2## is ther guarenteed to be a surface embedded in ##R^3##...
In Minkowski space, with line element $$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$$ (and ##c = 1##) we take spacelike trajectories to have ##ds^2 > 0##, null trajectories to have ##ds^2 = 0##, and timelike trajectories to have ##ds^2 < 0##. This makes sense given our definition of the line element...
Hi all:
As stated in the summary I'm in need for bibliography about timelike geodesics in the Kerr metric.
I have tried using the "Mathematical Theory of Black Holes" by S. Chandrasekhar but I find it a bit to complex.
Is there any other good books or articles about this that you might know...
(I am not sure which forum this post belongs to. Hope someone kindly helps me move it to a proper forum.)
In papers, for example, here, here, and here, the authors start from the Lagrangian for matters and gravitational fields, then Dirac's constrained canonical quantization is used. They...
Hi,
Just curious as to whether distances 'd' , used in Knn ; K nearest neighbors, in Machine Learning, are required to be metrics in the Mathematical Sense, i.e., if they are required to satisfy, in a space A:
##d: A \times A \rightarrow \mathbb R^{+} \cup \{0\} ;
d(a,a)=0 ;
d(a,b)=d(b,a) ...
Hi everyone! I'm having some difficulty showing that the variation of the four-velocity,
Uμ=dxμ/dτ
with respect the metric tensor gαβ is
δUμ=1/2 UμδgαβUαUβ
Does anyone have any suggestion?
Cheers,
Rafael.
PD: Thanks in advances for your answers; this is my first post! I think ill be...
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...
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...
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...
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...