What is Connection: Definition and 517 Discussions
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames. The connection form generally depends on a choice of a coordinate frame, and so is not a tensorial object. Various generalizations and reinterpretations of the connection form were formulated subsequent to Cartan's initial work. In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it is a differential form defined on the differentiable manifold, rather than on an abstract principal bundle over it. Hence, despite their lack of tensoriality, connection forms continue to be used because of the relative ease of performing calculations with them. In physics, connection forms are also used broadly in the context of gauge theory, through the gauge covariant derivative.
A connection form associates to each basis of a vector bundle a matrix of differential forms. The connection form is not tensorial because under a change of basis, the connection form transforms in a manner that involves the exterior derivative of the transition functions, in much the same way as the Christoffel symbols for the Levi-Civita connection. The main tensorial invariant of a connection form is its curvature form. In the presence of a solder form identifying the vector bundle with the tangent bundle, there is an additional invariant: the torsion form. In many cases, connection forms are considered on vector bundles with additional structure: that of a fiber bundle with a structure group.
I have read that we can define covariant derivative for spinors using the spin connection. But I can't see its proof in any textbook. Can anyone point to a reference where it is proved that such a definition indeed transforms covariantly ?
My month old laptop has recently developed frequent USB glitches. My mouse and audio interface connect via USB. About once a minute the connection fails for a few seconds.
I tried batteries in the mouse. Worse. I tried plugging the mouse and keyboard interface into the second USB port...
Hello,
I have a wifi device that has been connecting with no issues to my local wifi: it first detected the local wifi and I then used the password to complete the connection....
However, one day recently, the device disconnected from the wifi. After some reading, it seems that either low wifi...
Hi, Penrose in his book "The Road to Reality" claims that Newton/Galilean spacetime has actually a structure of fiber bundle. The base is one-dimensional Euclidean space (time) and each fiber is a copy of ##\mathbb E^3##. The projection on the base space is the "universal time mapping" that...
We had a thread long time ago concerning the Lie dragging of a vector field ##X## along a given vector field ##V## compared to the Fermi-Walker transport of ##X## along a curve ##C## through a point ##P## that is the integral curve of the vector field ##V## passing through that point.
We said...
@Greg Bernhardt
I've been getting a lot of connection timed out errors. Anything special going on at PF? Still using Windows 7. This is just satarted happening today.
quantum connection vs. quantum entanglement
Do they mean the same?
Are there any differences?
Is the second one more common/newer/more standard than the first one?
Could someone please answer these questions or recommend something to read?
If we consider the coefficient b as the rings impedance, we can consider the effective impedance on the right to be b+Z2 where Z2 is the impedance of the second string. Then because there is no reflection it follows that Z1=b+Z2 or b=Z1-Z2.
Is this a valid solution? My professor went through a...
I have 100 random real (even rational with only one decimal, like average temperatures of months at a particular weather station) numbers. With them I compute the arithmetical mean and the median. It is a (very) small probability they are the same number within let's say 0,1 or 0,2.
Question...
Hi, if the force is the derivative of potential energy, does it mean that the force is equal to mg and with a constant gravity, it will be the same at any height?
But in real life, F (or mg) would be different on the Earth's surface and 400 km above it (~8 m/s^2).
So, this formula is used to...
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...
I am facing this issue since some months on my laptop running Ubuntu 20.04. The WiFi disconnects suddenly and cannot reconnect, and an error message is shown in the Notification saying "Activation of Network Connection failed." This is not a problem with the router because just a few days back...
I don't understand why the highlighted term is there.
This image was taken from Sean Carroll's notes available here: preposterousuniverse.com/wp-content/uploads/grnotes-three.pdf
[Mentors' note: moved from technical forums so no template]
Hi All,
Working on a lab write-up, and I need background equations to support the reasoning for my experiment.
To outline briefly, two-part experiment, first part was finding the ideal pressure for a basketball, where I inflated it...
This isn't technically a homework problem, but I'm trying to check my understanding of the geometric phase by explicitly calculating the Berry connection for a simple 2x2 Hamiltonian that is not a textbook example of a spin-1/2 particle in a three dimensional magnetic field solved via a Bloch...
The answer from my teacher is two of these photovoltaic cells are connected in parallel. I can understand the parallel connection will produce same e.m.f but I don't understand why the total current will be greater than ##I##.
I imagined one of the cell connected to a lamp. The current through...
German Wikipedia mentions that the g factor is one of the causes of the Lamb Shift. It does not say why and I am trying to find a connection between these two things. Any ideas?
Hello guys,
I am supplying a Venturi Vac to my clients which is installed directly on their CNC Machine Coolant Pumps (photo attached).
This is done by introducing a Tee connection on the discharge end of the pump as shown. The straight line of the tee is connected to the hydraulic coolant...
While in QFT we remove infinite energy problem with renormalization procedure, asking e.g. "what is mean energy density in given distance from charged particle", electric filed alone would say $$\rho \propto |E|^2 \propto 1/r^4 $$
But such energy density would integrate to infinity due to...
I read somewhere that Morse originally applied his theory to the calculus of variations. I'm wondering, is this application useful in physics and mechanics, like maybe it sheds light on lagrangian mechanics? Does anyone know?
I am taking a course on General Relativity. Recently, I was given the following homework assignment, which reads
> Derive the following transformation rules for vielbein and spin connection:
$$\delta e_a^\mu=(\lambda^\nu\partial_\nu e_a^\mu-e_a^\nu\partial_\nu\lambda^\mu)+\lambda_a^b e_b^\mu$$...
Hi All,
I would like to know how can one connect the two definitions of entropy
##\Delta S = \int_{T_i}^{T_f} \frac{dQ}{T} ## and ##\Delta S = k_B \ln (\frac{W_f}{W_i})##,
particularly I am interested in how the logarithm emerges. Does it have to do with some linear dependence of the heat with...
Hello,
In a piping system connected to the outlet of an crude oil tank, process man has located the expansion joint after the valve which is connected to tank's nozzle. Since this expansion joint is located after the valve, it must be able to withstand the design pressure of the piping system...
Dear friends
i am using tensile test machine, tinius olsen 602 to conduct a tensile test for Glass-fiber composite. i am facing problem of using an extensometer from Epsilon (gage length is 25 mm). I have connected the extensometer to the tensile test machine(tinius olsen 602) through the...
Hi, I saw this video by numberphile, and near the end they mention how at the point of a right angle the equation shows infinite velocity for fluids. I'm wondering if this isn't perhaps related to Cantor's solution to Zeno's Paradox of distance (there's always a midpoint). Because I feel like at...
On wikipedia, it says that the Langlands program is a kind of "grand unified theory of mathematics." Does this program have any known connections to physics and in particular, quantum gravity?
Hi All, this is a steel frame , top beams sitting on top of column with steel plate cantilever at both sides (one side longer than the other) , so there will be hogging moment at the center support. However, the top steel beam is one whole piece, does the connection between the steel beam and...
I was trying to compute the equations of motion for the spin ##\omega_{\mu ab} ## connection from the Einstein-Palatini action
$$S := \int d^4 x e^\mu{}_ae^\nu{}_b R_{\mu \nu}{}^{ab}.$$
I managed to get the variation wrt. the connection to the form
$$ \varepsilon_{abcd}\varepsilon^{\mu \nu...
I am studying GR and I have these two following unresolved questions up until now:
The first one concerns the Levi-Civita connection. There are two conditions which determine the affine connections. The first one is that the connection is torsion-free (which is true for space-time and comes...
I recently came across a paper (referenced below) containing the statement that:"The differential form notation is much more concise and elegant than the tensor notation, but both contain the same information.", and the paper left me with a desire to understand the notation of differential...
I'm faced with a puzzle. Lately my desktop PC requires me to first disconnect, and then reconnect its mains cable, before it will start. By this I mean the cable has to be physically disconnected from the mains socket, as well as disconnected from the male plug that goes into the back of the...
In a parallel connection heat is produced.
R1 = 5 Ohm
R2 = 10 Ohm
What is the relation of W1/W2?
1:4
1:2
2:1
1:8
3:1
I’d tend to say 1:2, but I am not really sure…
Often I read that the Riemann Hypothesis (RH) is related to prime numbers because of the equivalence on Re(s)>1 of the zeta function and Eurler's product formula
, but is it more accurate that the relevance of the RH to primes (or vice-versa) is either that the RH implies formulas for the...
Hi,
I've been watching lectures from XylyXylyX on YouTube. I believe they are really great !
One doubt about the introduction of Covariant Derivative. At minute 54:00 he explains why covariant derivative is a (1,1) tensor: basically he takes the limit of a fraction in which the numerator is a...
I Think it's moment connection. Because the end plate must be able to resist the moment transferred from the beam to the column. Please correct me if I am wrong.
The following is a five-year-old post on one of NASA's websites.
https://www.nasaspaceflight.com/2015/04/evaluating-nasas-futuristic-em-drive/?fbclid=IwAR02ifoAVrPGa0StAxsTjIuS3b08fUOyMcPh1-nBjeiwgwWkI3H7jnA0Has
It exposes the holes in the state of affairs at that moment -- first, the article...
Lats say that i am moving on spaceship that moves 50% of the speed of light
And this spaceship is on a planet that moves in speed of 50% of the speed of light . And we have one clock on the spaceship ,clock number one, and on the planet we another clock,clock number two. And on difference...
If this is opening a can of worms then please say so and I'll start a separate thread.
This constraint on the covariant derivative means that transporting the metric is, in fact, path independent. Is this actually a requirement for any physically plausible theory? If you don't impose it...
Given a Weyl transformation of the metric ##g_{\mu\nu} \rightarrow g'_{\mu\nu} = e^{\Omega(x)} g_{\mu\nu}##, I'm trying to find the corresponding connection ##\Gamma'^{\lambda}_{\mu\nu}##, and from that ##-## via the Riemann tensor ##R'^{\lambda}_{\mu\nu\kappa}## ##-## the Ricci tensor...
Hi, just looking at RDP connections
https://support.microsoft.com/en-us/help/4028379/windows-10-how-to-use-remote-desktop
It would seem the two devices would have to be within the same LAN, right? Otherwise, how is just the computer name enough information to set up a connection? It seems for...
Hi there
I'm trying to access a FTP server in Windows Explorer. Getting to it, by inputting a username and password, works in Firefox but I'm needing to copy and paste a folder from the server to my desktop.
I get the following message when trying to access the server in Windows Explorer...
I am trying to derive the expression in components for the covariant derivative of a covector (a 1-form), i.e the Connection symbols for covectors.
What people usually do is
take the covariant derivative of the covector acting on a vector, the result being a scalar
Invoke a product rule to...
Hi All.
I hope this question makes sense.
In the case of Fourier Transforms one has the complex exponentials exp(2..π i. ξ.x)
In 3-D, if we single out where the complex exponentials are equal to 1 (zero phase), which is when ξ.x is an integer, a given ( ξ1,ξ2,ξ3).deﬁnes a family ξ.x= integer...
For as long as I can remember, the typical speaker connection on computer motherboards has 4 pins and the typical speaker has a 4 slot female connector with only two wires leading to the speaker. Is a there a technical or historical reason for the 4 pin connector?
So I am stumped by why these capacitors are there, to my understanding the capacitor C3 acts as a low pass filter but then I don't quite understand the purpose of C1 and C2 as indicated on the picture.
Many people talking about there are similarities and common positions in quantum entaglement and superposition with entropy. I need to know about this phenomenon