Tensor Definition and 1000 Threads

I was reading zee's group theory in a nutshell. I understand that we can decompose a 2 index tensor for rotation group into an antisymmetric vector(3), symmetric traceless tensor(5) and a scalar(trace of the tensor). Because "trace is invariant" it put a condition on the transformation of...

Background and Motivation The stress energy tensor of general relativity, as conventionally defined, has sixteen components. One of those component, conventionally component T00, also called ρ, is mass-energy density, including the E=mc2 conversion for electromagnetic fields. The other...

Hello all, let's suppose we have, in a flat spacetime, two observers O and O', the latter speeding away from O, with an uniform acceleration ##a##. In the Minkowski spacetime chart of O, the world-line of O' can be drawn as a parable. We know that the Lorentz boost at every point of the...

I know the tensor can be written as $$T^{\mu v}=\Pi^{\mu}\partial^v-g^{\mu v}\mathcal{L}$$ where $$g^{\mu v}$$ is the metric and $$\mathcal{L}$$ is the Lagrangian density, but how would I write $$T_{\mu v}$$? Would it simply be $$T_{\mu v}=g_{\mu \rho}g_{v p}T^{\rho p}$$? And if so, is there a...

I'm puzzling over Exercise 1.14 in Thorne & Blandford's Modern Classical Physics. We are given that an electric field ##\boldsymbol{E}## exerts a pressure ## \epsilon_{0}\boldsymbol{E}^{2}/2## orthogonal to itself and a tension of the same magnitude along itself. (The magnetic field does the...

I have an equation $$ \chi_\nu\nabla_\mu\chi_\sigma+\chi_\sigma\nabla_\nu\chi_\mu+\chi_\mu\nabla_\sigma\chi_\nu=0 $$so we also have$$ g_{\nu\rho}g_{\mu\tau}g_{\sigma\lambda}\left(\chi^\rho\nabla^\tau\chi^\lambda+\chi^\lambda\nabla^\rho\chi^\tau+\chi^\tau\nabla^\lambda\chi^\rho\right)=0 $$Does...

Given the action ##S =-\sum m_q \int \sqrt{g_{\mu\nu}[x_q(\lambda)]\dot{x}^\mu_q(\lambda)\dot{x}^\nu_q(\lambda)} d\lambda## The Energy-Momentum Tensor (EMT) is defined by the variation of the metric $$\delta S = \frac{1}{2}\int T_{\mu\nu} \delta g^{\mu\nu} \sqrt{g} d^4x$$ Then I use two...

If I have a matrix representing a 2nd order tensor (2 2) and I want to convert this matrix from M$$\textsuperscript{ab}$$ to $$M\textsubscript{b}\textsuperscript{a}$$ what do I do? I'm given the matrix elements for the 2x2 tensor. When applying the metric tensor to this matrix I understand...

For instance, using the vector ##A^\alpha e_\alpha##: ##g_{\mu \nu} e^\mu \otimes e^\nu (A^\alpha e_\alpha) = g_{\mu \nu} (e^\mu, A^\alpha e_\alpha) e^\nu ## ##g_{\mu \nu} e^\mu \otimes e^\nu (A^\alpha e_\alpha) = A^\alpha g_{\mu \nu} \delta_\alpha^\mu e^\nu = A^\mu g_{\mu \nu} e^\nu = A_\nu...

A second-order tensor is comprised at least of a two-dimensional matrix, as an nth-order tensor is comprised at least of an n-dimensional matrix, but what else is in the formal definition. A scientific definition needs to name the term being defined, and describe the meaning of that term only...

Can we consider the E and B fields as being irreducible representations under the rotations group SO(3) even though they are part of the same (0,2) tensor? Of course under boosts they transform into each other are not irreducible under this action. I would like to know if there is in some...

In a certain anisotropic conductive material, the relationship between the current density ##\vec j## and the electric field ##\vec E## is given by: ##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)## where ##\vec n## is a constant unit vector. i) Calculate the angle between the...

I know that a tensor can be seen as a linear function. I know that the tensor product of three spaces can be seen as a multilinear map satisfying distributivity by addition and associativity in multiplication by a scalar.

My idea is this: tensor stress is directly related to the internal pressure of a solid. That is to the force that the neighboring atoms exert each other in relation to a unit of surface. When I heat a solid we can have the phenomenon of thermal expansion: this is connected to the fact that a...

I know what Carroll refers to as 'conservation of indices' is just a trick to help you remember the pattern for transforming upper and lower components, but nonetheless I don't understand what he means in this example: E.g. on the LHS the free index ##\nu'## is a lower index, and on the RHS...

My try: $$ \begin{align*} \nabla^a T_{ab} &= \nabla^a \left(\nabla_{a} \phi \nabla_{b} \phi-\frac{C}{2} g_{a b} \nabla_{c} \phi \nabla^{c} \phi\right)\\ &\overset{(1)}{=} \underbrace{(\nabla^a\nabla_{a} \phi)}_{=0} \nabla_{b} \phi + \nabla_{a} \phi (\nabla^a\nabla_{b} \phi)-\frac{C}{2}...

Hello, I try to understand how to get the last relation below ##(3)## ( from stress energy tensor in special relativity - Wikipedia ). I understand how to get equation ##(1)## but I don't grasp how to make appear the gradient operator in the dot product and the divergence operator in the...

Hello, I'm trying to figure out where the term (3) came from. This is from a textbook which doesn't explain how they do it. ∂_μ(∂L/(∂(∂_μA_ν)) = ∂L/∂A_ν (1) L = -(1/16*pi) * ( ∂^(μ)A^(ν) - ∂^(ν)A^(μ))(∂_(μ)A_(ν) - ∂_(ν)A_(μ)) + 1/(8*pi) * (mc/hbar)^2* A^ν A_ν (2) Here is Eq (1) the...

$$\hat{T}_{\mu v}(x)=e^{i\hat{P}x}\hat{T}_{\mu v}(0)e^{-i\hat{P}x}$$, so $$\bra{\overrightarrow{P'}}\hat{T}_{\mu v}(x)\ket{\overrightarrow{P}}=e^{iP'x}\bra{\overrightarrow{P'}}\hat{T}_{\mu v}(0)\ket{\overrightarrow{P}}e^{-i\hat{P}x}$$ Now, $$\partial^{\mu}\Phi=\int\frac{d^3 k_1}{2\omega_{k_1}...

I have this statement: Find the most general form of the fourth rank isotropic tensor. In order to do so: - Perform rotations in ## \pi ## around any of the axes. Note that to maintain isotropy conditions some elements must necessarily be null. - Using rotations in ## \pi / 2 ## analyze the...

I've already made a post about this topic here, but I realized that I didn't understand the explanation on that post. in Chapter 7 of Rindler's book on relativity, in section about electromagnetic field tensor, he states that _and introducing a factor 1/c for later convenience, we can ‘guess’...

The textbook gives some examples for ##P_1 \left ( cos \theta \right)##, ##P_2 \left ( cos \theta \right)##, and ##P_3 \left ( cos \theta \right)##. The procedure is clear. The key to the problem is to find the symmetric traceless tensor for the proper rank. I first construct a symmetric...

Hi. I am looking for a book about tensor analysis. I am aware that there have been some post about those books, but I wish to find a thin book rather than a tome but just good enough for physics, such as group theory, relativistic quantum mechanics, and quantum field theory. I am reading...

I read in the book Gravitation by Wheeler that "Any tensor can be completely symmetrized or antisymmetrized with an appropriate linear combination of itself and it's transpose (see page 83; also this is an exercise on page 86 Exercise 3.12). And in Topology, Geometry and Physics by Michio...

I am afraid I had no credible attempt at solving the problem. My poor attempt was writing the matrix ##\mathbb R## as a ##3 \times 3## square matrix with elements ##a_{ij}## and use the matrix form of the orthogonality relation ##\mathbb R^T \mathbb R = \mathbb I##, where ##\mathbb I## is the...

Aside from being symmetric, are there any other mathematical constraints on the metric?

Where do I start. I want to write the matrix form of a single or two qubit gate in the tensor product vector space of a many qubit system. Ill outline a simple example: Both qubits, ##q_0## and ##q_1## start in the ground state, ##|0 \rangle =\begin{pmatrix}1 \\ 0 \end{pmatrix}##. Then we...

Let's say we have any two covariant derivative operators ##\nabla## and ##\nabla'##. Then there exists a tensor ##C^{\alpha}_{\mu\nu}## such that for all covariant vectors ##\omega_{\nu}##,$$\nabla_{\mu}\omega_{\nu}=\nabla'_{\mu}\omega_{\nu}-C^{\alpha}_{\mu\nu}\omega_{\alpha}$$ Now I'm quoting...

I want to compute the Riemann Tensor of the following metric $$ds^2 = dr^2+(r^2+b^2)d \theta^2 +(r^2+b^2)\sin^2 \theta d \phi^2 -dt^2$$ Before going through it I'd like to try to predict how many non-trivial components we'd expect to get, based on the Riemann tensor basic rule: It is...

About 10 years ago I worked on a project where I took a mater distribution and numerically solved for spatial curvature. Can this be done in the opposite direction? Can anybody point me to a resource that would allow me to calculate matter distributions when the metric is specified? What are...

There are a few different textbooks out there on differential geometry geared towards physics applications and also theoretical physics books which use a geometric approach. Yet they use different approaches sometimes. For example kip thrones book “modern classical physics” uses a tensor...

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...

Good day all. Given that in Sean Carroll`s Lectures on GR he states that when calculating the covariant derivative of a 1-Form the Christoffel symbols have a negative sign as opposed to for the covariant derivative of a vector, would it be naive to think that, given the usual equation for the...

Tong proposes the following exercise in this lecture (around 25:30, section b)): Exercise statement: Prove that the stress-energy tensor is given by the functional derivative of the action with respect to ##\delta g^{\mu \nu}## $$T_{\mu \nu} = \frac{-2}{\sqrt{-g}} \frac{\delta S}{\delta...

Prove that the stress-energy tensor is given by the functional derivative of the action with respect to ##\delta g^{\mu \nu}## $$T_{\mu \nu} = \frac{-2}{\sqrt{-g}} \frac{\delta S}{\delta g^{\mu \nu}}$$ Tong suggests (around 25:30) we could get the desired tensor by performing a...

We have 4-tensor of second rank. For example energy-momentum tensor ##T_μν## , which is symmetric and traceless. Then ##T_{μν}=x_μx_ν+x_νx_μ## where ##x_μ## is 4-vector. Every 4- vector transform under Lorentz transform as (12,12). If we act on ## T_{μν}## , by representation( with...

I created an array, where the first three entries of each column are the x,y, and z coordinates. The last entry in each column is the charge. I called this array PCQ. l/2 l/2 -l/2 -l/2 -l/2 l/2 l/2 -l/2 -l/2 l/2 l/2 -l/2 -l/2 -l/2 l/2 l/2 l/2 l/2 l/2 l/2 -l/2 -l/2 -l/2 -l/2 q -q q -q q...

I'm following this video on how to establish an equivalence relation to define the tensor product space of Hilbert spaces: ##\mathcal{H1} \otimes\mathcal{H2}={T}\big/{\sim}## The definition for the equivalence relation is given in the lecture vidoe as ##(\sum_{j=1}^{J}c_j\psi_j...

REMARK: First of all I have to say that this Lagrangian reminds me of the Lagrangian from which we can derive Maxwell's equations, which is (reference: Tong QFT lecture notes, equation 1.18; I have attached the PDF). $$\mathcal{L} = -\frac 1 2 (\partial_{\mu} A_{\nu} )(\partial^{\mu} A^{\nu}) +...

The stress energy tensor has many forms based on the type of matter you are describing, dust, fluid, perfect fluid... is it true that the trace of all of these matter situations is invariant?

Under the coordinate transformation $\bar x=x+\varepsilon$, the variation of the metric $g^{\mu\nu}$ is: $$ \delta g^{\mu\nu}(x)=\bar g^{\mu\nu}(x)-g^{\mu\nu}(x)=-\frac{\partial{ g^{\mu\nu}}}{\partial x^{\alpha}}\varepsilon^{\alpha}+ g^{\mu\beta}\frac{\partial \varepsilon^{\nu}}{\partial...

I am trying to answer exercise 5 but I am not sure I understand what the hint is implying, differentiate with respect to ##p_\alpha## and ##p_\beta##, I have done this but nothing is clicking. Also, what is the relevance of the hint "the constraint ##p^\alpha p_\alpha = m^2c^2## can be ignored...

I read recently that Einstein initially tried the Ricci tensor alone as the left hand side his field equation but the covariant derivative wasn't zero as the energy tensor was. What is the covariant derivative of the Ricci tensor if not zero?

I have read many GR books and many posts regarding the title of this post, but despite that, I still feel the need to clarify some things. Based on my understanding, the contravariant component of a vector transforms as, ##A'^\mu = [L]^\mu~ _\nu A^\nu## the covariant component of a vector...

Hello, I was wondering why the energy-stress tensor only accounts for electromagnetic Energy Density and does not include the other forces? Secondary question could this be a flaw within the mathematics of GR making it give nonsense answers for Quantum level interactions?

I need to use some property of the relalation between the coordinate systems to prove that g_{hk} is independent of the choice of the underlying rectangular coordinate system. I will try to borrow an idea from basic linear algebra. I expect any transformation between the rectangular systems to...

Suppose the Bell operator ##B=|AB(1,2)+AB(1,3)+AB(2,3)|## With ##AB\in{1,-1}## Nonlocal realism implies ##B\in{1,3}## However using usual matrix sum one eigenvalues for the result of measurement can be smaller than 1, implying nonlocal realism cannot explain the quantum result. However if...

In Special Relativity I'm given the energy-momentum tensor for a perfect fluid:$$ T^{\mu\nu}=\left(\rho+p\right)U^\mu U^\nu+p\eta^{\mu\nu} $$where ##\rho## is the energy density, ##p## is the pressure, ##U^\mu=\partial x^\mu/\partial\tau## is the four-velocity of the fluid. In the...

My attempt was to first rewrite ##S_M## slightly to make it more clear where ##g_{\mu\nu}## appears $$S_M = \int d^4x \sqrt{-g} (g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi-\frac{1}{2}m^2\phi^2).$$ Now we can apply the variation: $$\begin{align*} \delta S_M &= \int d^4x (\delta\sqrt{-g})...

When discussing how a rank two tensor transforms under SO(3), we say that the tensor can be decomposed into three irreducible parts, the anti-symmetric part, traceless-symmetric part, and a 1-dimensional trace part, which transforms as a scalar. How do we know that the symmetric and...