In mathematics, a tensor is an algebraic object that describes a (multilinear) relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include vectors and scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system.
Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), or general relativity (stress–energy tensor, curvature tensor, ...) and others. In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".
Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 - continuing the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others - as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.
I came across a statement in《A First Course in General Relativity》:“The only matrix diagonal in all frames is a multiple of the identity:all its diagonal terms are equal.”Why?I don’t remember this conclusion in linear algebra.The preceding part of this sentence is:Viscosity is a force parallel...
I came across a statement in《A First Course in General Relativity》on page 97 which confused me.It read:"if the forces are perpendicular to the interfaces,then##T^i{^j}##will be zero unless ##i=j##".
Where the ##T## is stress-energy tensor,##T^i{^j}##is the flux of i momentum across the j surface.
ep_{ijkl} M^{ij} N^{kl} + ep_{ijkl}N^{ij} M^{kl}
The second term can be rewritten with indices swapped
ep_{klij} N^{kl}M^{ij}
Shuffle indices around in epsilon
ep{klij} = ep{ijkl}
Therefore the expression becomes
2ep_{ijkl}M^{ij}N^{kl}
Not zero.
What is wrong here?
I want to get the stress energy tensor of a scalar field using the Hilbert method (namely, ##T^{\mu v} = \frac{2}{\sqrt{-g}} \frac{\delta S}{\delta g_{\mu v}}##)
$$S = \int \frac{1}{2}(\partial_\mu \phi \partial^{\mu} \phi - m^2 \phi ^2)\sqrt{-g}d^4x$$
$$= \int \frac{1}{2}(\partial^{v} \phi...
Let's arrange the rod's axis parallel to the z axis.
##T_{00} = A/\mu## (since it represents the energy density)
##T_{03}=T_{30} = \frac{F\sqrt{\mu / F}}{A}## (It represents the flow of energy across the z direction)
##T_{33} = F/A## (pressure)
It seems that ##T_{33}## i have got has the...
We mainly have to prove that this quantity## \bra{\varphi} A^{\otimes n } \ket{\varphi} \pm \bra{\varphi} B^{\otimes n } \ket{\varphi} ##
is greater or equal than zero for all ##\ket{\varphi}##.
Being ##\ket{\varphi}## a product state it is straightforward to demonstrate such inequality. I am...
Is tensor product the same as dyadic product of two vectors? And dyadic multiplication is just matrix multiplication? You have a column vector on the left and a row vector on the right and you just multiply them and that's it? We just create a matrix out of two vectors so we encode two...
I have the matrix
$$
A = \left(\begin{array}{cc}
y^2 & -xy\\
-xy & x^2
\end{array} \right)
$$
I know that to prove that the matrix is a tensor, it transform their elements in another base. But I still without how begin this problem.
Help please! Thanks.
hello,
1. according to Robert Wald, General Relativity, equation (4.2.22)
the magnetic field as measured by an observer with 4-velocity ## v^b ## is given by
## B_a = - \frac {1}{2} {ϵ_{ab}}^{cd} F_{cd} v^b ##
where ## {ϵ_{ab}}^{cd}##, the author says, is the totally antisymmetric tensor (for...
i am facing problems in the definition of dual oF some objects which has pair of anti symmetric indices e.g. Weyl curvature tensor. Double dual is there in the literature but given that how to find the anti self dual part of that. the problem is written in attached the file.
I am completely stuck on problem 2.45 of Blennow's book Mathematical Models for Physics and Engineering. @Orodruin It says
"We just stated that the moment of inertia tensor ##I_{ij}## satisfies the relation$${\dot{I}}_{ij}\omega_j=\varepsilon_{ijk}\omega_jI_{kl}\omega_l$$Show that this relation...
According to Wikipedia, the definition of the Riemann Tensor can be taken as ##R^{\rho}_{\sigma \mu \nu} = dx^{\rho}[\nabla_{\mu},\nabla_{\nu}]\partial_{\sigma}##. Note that I dropped the Lie Bracket term and used the commutator since I'm looking at calculating this w.r.t. the basis. I...
I am a beginner in GR, working my through Gravitation by the above authors. If there is a better place to ask this question, please let me know.
I understand (from section 5.7) that the stress-energy tensor is symmetric, and from equation 5.23 (p. 141), it is explicitly symmetric. But...
Galaxies are very large rotating bodies, so it seems that, as with the Kerr model for black holes, there could be an effect of this global rotation on the energy momentum tensor in the more dense regions of the galaxy that could in turn affect the space-time in the vicinity of the object and so...
I am in a course in applied strength of materials and we often use the 3D stress tensor for stress analysis of materials i.e. Mohr's circles, bending, torsion, etc. Is the stress-energy tensor in relativity basically a 4-d extension to the Cauchy stress tensor commonly used in mechanical...
When we compute the stress energy momentum tensor ## T_{\mu\nu} ##, it has units of energy density. If, therefore, we know the total energy ##E## of the system described by ## T_{\mu\nu} ##, can we compute the volume of the system from ## V = E/T_{00}##?
If it holds, I would assume this would...
How to write following equation in index notation?
$$\nabla \cdot \left( \mathbf{e} : \nabla_{s} \mathbf{u} \right)$$
where ##e## is a third rank tensor, ##u## is a vector, ##\nabla_{s}## is the symmetric part of the gradient operator, : is the double dot product.
The way I approached is...
If I have an equation, let's say,
$$\mathbf{A} = \mathbf{B} + \mathbf{C}^{Transpose} \cdot \left( \mathbf{D}^{-1} \mathbf{C} \right),$$
1.) How would I write using index notation? Here
A is a 4th rank tensor
B is a 4th rank tensor
C is a 3rd rank tensor
D is a 2nd rank tensor
I wrote it as...
Hello all,
I am hoping to get some feedback on the manner in which I performed computations towards solving the following problem.
There are a couple specific points which I am not confident of:
1. Did I properly account for the manifold structure in my computation of the nonzero components...
In a general coordinate system ##\{x^1,..., x^n\}##, the Covariant Gradient of a scalar field ##f:\mathbb{R}^n \rightarrow \mathbb{R}## is given by (using Einstein's notation)
##
\nabla f=\frac{\partial f}{\partial x^{i}} g^{i j} \mathbf{e}_{j}
##
I'm trying to prove that this covariant...
I'm still confused about the notation used for operations involving tensors.
Consider the following simple example:
$$\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...
In Gravitation by Misner, Thorne and Wheeler (p.139), stress-energy tensor for a single type of particles with uniform mass m and uniform momentum p (and E = p2 +m2) ½ ) can be written as a product of two 4-vectors,T(E,p) = (E,p)×(E,p)/[V(E2 – p2 )½ ]
Since Einstein equation is G = 8πGT, I am...
What is the index notation of divergence of product of 4th rank tensor and second rank tensor?
What is the index notation of divergence of 3rd rank tensor and vector?
div(a:b) = div(c^transpose. d)
Where a = 4th rank tensor, b is second rank tensor, c is 3rd rank tensor and d is a vector.
It's given as ##T_{\mu \nu} = - \mathrm{tr}(F_{\mu \lambda} {F_{\nu}}^{\lambda} - \frac{1}{4} g_{\mu \nu} F_{\alpha \beta} F^{\alpha \beta})##. Can somebody explain the notation, i.e. what is the meaning here of the trace? (usually I would interpret the trace of a matrix as the number...
Hi all,
I am currently trying to prove formula 21 from the attached paper.
My work is as follows:
If anyone can point out where I went wrong I would greatly appreciate it! Thanks.
I've started reading up on tensors. Since this lies well outside my usual area, I need some clarifications on some tensor calculus issues.
Let ##A## be a tensor of order ##j > 1##. Suppose that the tensor is cubical, i.e., every mode is of the same size. So for example, if ##A## is of order 3...
Goldstein pg 192, 2 edIn a Cartesian three-dimensional space, a tensor ##\mathrm{T}## of the ##N## th rank may be defined for our purposes as a quantity having ##3^{N}## components ##T_{i j k}##.. (with ##N## indices) that transform under an orthogonal transformation of coordinates...
Hi,
I would like to ask for a clarification about the difference between a differential k-form and a generic (0,k) tensor field.
Take for instance a (non simple) differential 2-form defined on a 2D differential manifold with coordinates ##\{x^{\mu}\}##. It can be assigned as linear combination...
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...
Let ##\varphi## be some scalar field. In "The Classical Theory of Fields" by Landau it is claimed that
$$
\frac{\partial\varphi}{\partial x_i} = g^{ik} \frac{\partial \varphi}{\partial x^k}
$$
I wanted to prove this identity. Using the chain rule
$$
\frac{\partial}{\partial x_{i}}=\frac{\partial...
I need to vary w.r.t ##a_{\alpha \beta} ##
##\frac{\partial L}{\partial_{\mu}(\partial_{\mu}{a_{\alpha\beta}})}-\frac{\partial L}{\partial {a_{\alpha \beta}}}## (1)
I am looking at varying the term in the Lagrangian of ##\frac{1}{3}A^{\mu} \partial_{\mu}\Phi ##
where ##A^{\beta}=\partial_k...
Hello
I am reviewing the proof of Cauchy's formula for the stress tensor and surface traction.
Without exception, every book I look at gets to the critical point of USING the projection of a triangle onto one of the three orthogonal planes.
However, I have never seen this proven.
I have...
According to my book, the equation that should meet a vector ##\mathbf{v}=v^i\mathbf{e}_i## in order to be parallel-transported in a manifold is:
##v_{, j}^{i}+v^{k} \Gamma_{k j}^{i}=0##
Where ##v_{, j}^i## stands for ##\partial{v^i}{\partial y^j}##, that is, the partial derivative of the...
The covariant form for the Levi-Civita is defined as ##\varepsilon_{i,j,k}:=\sqrt{g}\epsilon_{i,j,k}##. I want to show from this definition that it's contravariant form is given by ##\varepsilon^{i,j,k}=\frac{1}{\sqrt{g}}\epsilon^{i,j,k}##.My attemptWhat I have tried is to express this tensor...
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...
I would like to know what is the utility or purpose for which the elements below were defined in the Tensor Calculus. They are things that I think I understand how they work, but whose purpose I do not see clearly, so I would appreciate if someone could give me some clue about it.
Tensors. As...
I'm trying to understand why it is possible to express vectors ##\mathbf{e}^i## of the dual basis in terms of the vectors ##\mathbf{e}_j## of the original basis through the dual metric tensor ##g^{ij}##, and vice versa, in these ways:
##\mathbf{e}^i=g^{ij}\mathbf{e}_j##...
Property (a) simply states that a second rank tensor that vanishes in one frame vanishes in all frames related by rotations.
I am supposed to prove: ##T_{i_1 i_2} - T_{i_2 i_1} = 0 \implies T_{i_1 i_2}' - T_{i_2 i_1}' = 0##
Here's my solution. Consider,
$$T_{i_1 i_2}' - T_{i_2 i_1}' = r_{i_1...
Can the energy-momentum tensor of matter and energy be cast in terms of energy density of matter and energy, similar to how the energy-momentum tensor of vacuum energy can be cast in terms of the energy density of vacuum energy?
> **Exercise.** Let T1and T2be tensors of type (r1 s1)and (r2 s2) respectively on a vector space V. Show that T1⊗
T2can be viewed as an (r1+r2 s1+s2)tensor, so that the
> tensor product of two tensors is again a tensor, justifying the
> nomenclature...
What I’m reading:《An introduction to...
I'm looking for literature recommendations regarding tensor networks. I never came across singular value decomposition or spectral decomposition in my linear algebra classes, so I need to brush up on the relevant mathematical background as well.
Hello everyone,
in equation 3.86 of this online version of Carroll´s lecture notes on general relativity (https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html) the covariant derviative of the Riemann tensor is simply given by the partial derivative, the terms carrying the...
In "Gravitational radiation from point masses", by Peters & Mathews, http://gravity.psu.edu/numrel/jclub/jc/Peters_Mathews_PR_131_435_1963.pdf, the emitted power from gravitatioanal quadrupole radiation per unit solid angle ##\Omega## is given by:
$$ \frac{dP}{d\Omega} = = \frac{ G} {8 \pi c^2...