Scalar Definition and 829 Threads

I have been searching for a way to relate known concepts (known to me) to the computation of the dot product in an effort to understand why it takes the form it does. I ran into a little snippet in a classical dynamics book that seems like it just may be the ticket. Here is what it says...

Homework Statement I'm currently self-studying Carroll's GR book and get stuck by proving the following identity: K^\lambda \nabla _\lambda R = 0 where K is Killing vector and R is the Ricci ScalarHomework Equations Mr.Carroll said that it is suffice to show this by knowing: \nabla _\mu...

The Kretschmann curvature scalar is defined to be K = RabcdRabcd where Rabcd is the Riemann tensor. I believe I heard in class that this scalar can be used to demonstrate the existence of a curvature singularity. Can somebody tell me why this is so? Also, I heard that it is better (easier I...

i have vector "a" why a*a=1?? it make no sense the formula says |a|*|a|*cos0=a^2 (not 1)

Hi, I got a doubt question for this. Given this general expression for the scalar function, G such that del G = F(x,y,z)=2xyi + (x^2 - Z^3)j + (-3yz^2 + 1)k How do i go about finding G?

Why is the Higgs field a scalar field? I understand if it is one, it will have no spin and no angular momentum. But understanding that a particle is a scalar seems to me a leap of faith. What am I not getting?

Greetings all, I'm reading about a way to solve for the volume of a "parallelepiped" in 3 space, which is determined by vectors u, v and w. The volume is apparently the absolute value of the determinant given by the matrix u1 u2 u3 v1 v2 v3 w1 w2 w3 which is the same as the scalar triple...

Homework Statement I have a question to find Scalar factors of parabolic cylindrical coords and element dV with provided tranformation equations. I know the values for both of them and that the product of the scalar factors is the dV, but how do i derive those scalar factors? I don't even know...

I don't understand how a scalar can be negative. Like work for example, how can this be - or + yet still be a scalar. I've read that scalars only have magnitude, while vectors have both magnitude and direction. As vague as these definitions seem, I reason that scalars wouldn't have any +/-...

Okay I know the definition of a Vector and of a scalar... but I am getting a little confused for some reason. Wolfram.com gives this definition of a scalar function: A function f(x_1,x_2,...,x_n) of one or more variables whose range is one-dimensional, as compared to a vector function...

Homework Statement Find the commutators [P^\sigma,J^{\mu \nu}] The answer is part of the Poincare algebra [P^\sigma,J^{\mu \nu}]=i(g^{\mu \sigma}P^\nu-g^{\nu \sigma}P^\mu) If someone can convince me that \partial_i T^{0\mu} = 0, (i.e. the energy-momentum tensor has no explicit spatial...

Homework Statement Working on an exercise from Srednicki's QFT and something is not clear. Show that [\varphi(x), M^{uv}] = \mathcal{L}^{uv} \varphi(x) where \mathcal{L}^{uv} = \frac{\hbar}{i} (x^u \partial^v - x^v \partial^u ) Homework Equations (1) U(\Lambda)^{-1} \varphi(x)...

The action of non-spinning particles in scalar gravity is S=-\int{\sqrt{-g}(\frac{1}{8\pi}g^{\mu\nu}\Phi_{,\mu}\Phi_{,\nu}-\rho e^{\Phi})d^4x} where \rho presents the comoving density. Now, I want to know the formula when particles with classical spin. Thank you!

Hey, in my textbook they keep doing this and I can't follow for example r.\ddot{}r = 1/2 \ddot{}r^{}^2{} and r.\dot{}r = 1/2 \dot{}r^2{}. Can anyone explain this to me? I know I should probably know it. P.S Can't quite get the dot product to look right apologies.

I was trying to expand a scalar function with a power series but it accepts a vector argument. Do I simply use the multivariable power series expansion with the components of the vector acting as the argument OR do I use the single variable power series and take the vector's magnitude in the...

I have a question on an assignment that expects the I+J form of a vector but is only giving the direction(cardinal) or angle of the vector. See example below: Homework Statement Unit vectors and are directed east and north, respectively. Calculate the unit vector (in terms of I and J) in...

Homework Statement I am confused how the scalar multiple is divided out of the proof of this rule without taking an h with it in the denominator, which would get very tiny meaning the entire thing would go to infinity or negative infinity or zero, you can't tell. Start with: f(x) = k...

a) displacement b) velocity c) average velocity d) distance e) speed f) acceleration g) position which of these are vector and which are scalar?

The two dimensional action is: S_k = \int d^2\sigma\sqrt{h}\left(\partial_\alpha\phi\partial^\alpha\phi - \frac{i}{2}kR^{(2)}\phi\right) where k is a constant, R^{(2)} is the two dimensional scalar curvature. I'm trying to derive the following energy momentum tensor: T_{\alpha\beta}^k =...

Actually, I'm not fully understand what the meaning of conformal dimension is. But I know how to read off the conformal dimension of a tensor, say, t^{++}{}_+, then the conformal dimension is -2 + 1= -1, where the lower index carries conformal dimension 1 and upper index carries conformal...

I have a question concerning scalar invariance with respect to an accelerating and an inertial reference frame. Here is the problem. Suppose we have a rotating spherical object, which we denote as the rotator, attached to a near-massless wire. The other end of the wire slips loosely over a...

(Sorry, the title should read "...why curl of gradient of a scalar "function" is zero) Of course I know how to compute curl, graident, divergence. Algebrically I know curl of gradient of a scalar function is zero. But I want to know the reason behind this...and also the reason why gradient of...

I'm re-visiting calculus again, and I've stumbled onto the concepts of scalar and vector projections in the vector chapter. While keeping in mind which equation to use for what projection is quite doable, I cannot seem to see the purpose of keeping scalar and vector projections in mind. Can...

Is there a general way of proving that the scalar product xuxu = (x0)2 - (x1)2 - (x2)2 - (x3)2 is invariant under a Lorentz transformation that applies no matter the explicit form of the transformation.

Are they real are what? I've read in various places on the internet that there is a vast amount of energy stored in the vacuum of space that is supposedly extractable and extremely efficient. The top link below contains quite a bit on information about the potential weapons (scalar howitzer...

"A scalar field assigns every point in space to a scalar value" Would this be a correct definition of a scalar field? Thanks

As I understand it, for a tensor of any rank I can produce a corresponding scalar in the following way: Create an inverted form of the tensor by lowering its upper indices and raising its lower indices, and then taking the inner product of this tensor and the original one. My only question...

Homework Statement Use the scalar projection to show that a distance from a point P(x1, y1) to the line ax + by + c = 0 is \frac{ax1 + by1 + c}{\sqrt{a^2 + b^2}}Homework Equations scalar projection = \frac{a . b}{|a|} The Attempt at a Solution The first thing that I did was to say that b =...

Hi I have a small subtle problem with the sign of the energy-momentum tensor for a scalar field as derived by varying the metric (s.b.). I would appreciate very much if somebody could help me on my specific issue. Let me describe the problem in more detail: I conform to the sign convention...

Ok this is a pretty simple concept and in the electronics field I use it on a monthly basis, I can do it but I do not understand how it works and It is bugging the crap out of me. EXAMPLE: I have a pressure sensor, it reads pressure from 0 psi to .73 psi, and outputs a corresponding voltage...

Logic of E-H action, ricci scalar, cosmological constant?? This crazy thread is mean to stimulate some reflections on the logic of Einsteins Equations. It would be interesting if those who have any ideas can join. Maybe it could be enlightning? The common way of thinking about GR is that we...

Many of you will have heard of these. Does anyone knowledgeable on conventional electromagnetics, suspect there may be some truth in it?

I have seen in one paper that photon is coupled to dilaton field which is scalar and motivated by string theory. I do not understand this. Photon is carrier of electromagnetic field and so I thought it can only couple to electrically charged fields. Can anyone explain?

Suppose I couple a fermion field to a scalar field using \mathrm{i} g \bar{\psi}\psi \varphi and \mathrm{i} g \bar{\psi}\gamma_5\psi\varphi. I'm trying to understand what would be the physical difference between these interactions. I know that (1/2)(1\pm \gamma_5) approximately projects out...

What is the meaning of the curvature scalar (R) in GR? More precisely, what is the meaning of it's evolution? Why when we are concerning the solar system we take R to be small and when we are concerning the cosmological scales the we assume R to be large? Thanks in advance.

I've been trying to work my way through some of my lecture notes, and have hit this snag. (n.b. I use k_0 \equiv +\sqrt{\vec{k}^2 + m^2}) We have a(q) = \int d^3 x e^{iqx} \{ q_0 \phi(x) + i \pi(x) \} a^{\dagger}(q) = \int d^3 x e^{-iqx} \{ q_0 \phi(x) - i \pi(x) \} To calculate the...

Hi guys and gals This is a conceptual question. Let's say I have a scalar function, f(x,y,z) defined throughout \mathbb{R}^3. Further I have some bounded surface, S embedded in \mathbb{R}^3. How would I find the function f, defined on the surface S? Would it be the inner product of f...

Does it make sense for the Ricci Scalar to be a function of the spacetime coordinates? In previous calculations I have carried out in the past, everytime the Ricci Scalar has been returned as a constant, rather than being explicitly dependent on the coordinates. Thanks for any replies

Attached is a .jpg of my problem. I know how to find the scalar product of B*C (I think... 5, right?), but I don't really know where the 2 and 3 come into play. I've tried multiplying the values of C by 3 and then finding the scalar product, then multiplying the quantity by two, but that was...

Suppose you are given the Lagrangian of a scalar field \Phi(t) \mathcal{L} = \frac{1}{2} \dot{\Phi}- \nabla \Phi - V(\Phi ). By introducing covariant notation with \eta_{\mu \nu} = (1,-1,-1,-1) this reads as \mathcal{L} = \frac{1}{2} \eta^{\mu \nu} \partial_\mu\Phi \;\partial_\nu\Phi-...

Hello all. In a quite easy to follow short piece by Edmond Bertschinger entitled Introductio to Tensor Calculus for General Relativity on page 6 when speaking of the metric tensor he says, referring to the symbol conventions used in the piece :- "" We reserve the dot product notation for...

I am given the symmetric equation of the line of intersection of two planes. I also have a point on each plane. Now I have to work backwards to determine the scalar equations of the two planes. I can work to get it by saying for example x + 2y + 3z -6 + k(4x - y -z +4) - these are made up...

I'm kind of confused as to how to determine when a vector field is conservative. For example, if we consider the following scalar field: \varphi = arctan(\frac{y}{x}) We see that the gradient is: F = \nabla\varphi = \frac{-y i + x j}{x^{2} + y^{2}} However, F is not a conservative...

Hey, this is rather involved but I hope someone can help me out. I am reading http://arxiv.org/abs/0704.3626 ( the casimir force in randall sundrum models) and am trying to get from equation 2.1 : g^{\mu\nu}\partial_\mu\partial_\nu\Phi+e^{2ky}\partial_y(e^{-4ky}\partial_y\Phi)=0 to...

I see in several physics papers and articles current, current density represented by vectors. But in one book it was mentioned clearly that current is not a vector because it does not obey vector law of addition. Can some one clarify this point with an example and show that current does not...

Hello there When multiplying two vectors, why do we multiply them with the cosine of the angle between them? why not the sine of the angle? Thanks.

Hi all, I have a little problem concerning the coupling of a fermion to CP^N (or better a 2D scalar O(3) model). Its not a mathematical type of problem. I just read on "The coupling of fermions to the three-dimensional noncommutative $CP^{N-1}$ model: minimal and supersymmetric extensions"...

I've been wondering about terms you typically find in the action of a field theory, for example consider the kinetic term of a scalar field S=\int d^4x(\partial_\mu\phi\partial^\mu\phi). I've read that it can be thought of as the kinetic energy of the field - but this just doesn't sit to...

How can I write the most general real scalar potential in terms of the singlets and doublets corresponding to irreducible representations of S3?

"Consider a surface S on which a scalar field f is defined" "Consider a surface S on which a scalar field f is defined" what does "on which is defined" mean phys descript answers appreciated!