prove a tensor is symmetric

I'm not sure if this is only for SR or if also for GR since we've only been talking about SR thus far, though GR is something we'll be covering soon. I'm completely lost on this. Example with the metric tensor we have distance between 2 points as being invariant under coordinate transformation. decompositions of the input symmetric tensor . Following the definition of a vector, we can define a $(2,0)$ tensor $T$ (not necessarily a product of vectors as above) as a function that assigns a set of numbers $T^{ij}_x$ to each coordinate system, such that the components in two different systems follow the above transformation law. How to fight an unemployment tax bill that I do not owe in NY? $$\frac{d u_\mu}{d\tau} = (q/m) F_{\mu\nu} u^\nu.$$, $$\Delta u_\mu = \Lambda_{\mu\nu} u^\nu$$, http://www.math.ucla.edu/~baker/149.1.02w/handouts/e_htls.pdf, http://www.lecture-notes.co.uk/susskind/special-relativity/lecture-6/rank-two-tensors/, [Physics] Can we think of the EM tensor as an infinitesimal generator of Lorentz transformations, [Physics] Transformation of the Levi Civita symbol Carroll. (I am using subscripts to label coordinate systems.) It only takes a minute to sign up. In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold M (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. is also positive. Every Second Order Tensor Can be Expressed as The Sum of Symmetric and Anti Symmetric Tensor, Tensors as a Sum of Symmetric and Antisymmetric Tensors, Tensor 13 | Symmetric and Antisymmetric Tensor, symmetric and antisymmetric tensor (hindi), 2. I know if the transformation is for rotations then $\Lambda^{\mu}$ is just the orthogonal rotation matrix. The Levi-Civita tensor is a true tensor, which yields the volume spanned by an ordered set of basis vectors. How to clarify that supervisor writing a reference is not related to me even though we have the same last name? stress tensor), but is a more general concept that describes the linear relationships between objects, independent of the choice of coordinate system. A tensor is not particularly a concept related to relativity (see e.g. Because the stress tensor is uniform, the force is the same, but now the normal component $\tau_{22}\hat{e}^{(2)}$ exerts a clockwise torque, equal but opposite to that exerted by the same component at point A. We argue that stress components located above and below the main diagonal represent torques that are equal but opposite. Then you should obtain the same result. I suspect that the actual question is as follows. In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: for every permutation of the symbols {1, 2, ., r}. But let's just talk about a rank (2,0) symmetric contravariant tensor for a second, denoted $S^{\mu\nu}$ and equals $S^{\nu\mu}$. (NOTE: I don't want to see how these terms being symmetric and antisymmetric explains the expansion of a tensor. If you have a matrix $\Lambda^i_{\ j}$ relating coordinates $x$ and $y$ as above, it makes no sense to ask what $\Lambda$ looks like in coordiantes $z$. Do sandcastles kill more people than sharks? - These can be read about at: http://www.math.ucla.edu/~baker/149.1.02w/handouts/e_htls.pdf. Our book uses $R$ in place of $\Lambda$ in their formulations above, where $R$ might just be rotations. The indices arise by letting the tensor act on the basis vectors. }\right)^{T}, \bar{n}^{(1)^{T}} \cdot A =\bar{\lambda}_{1} \bar{n}^{(1)^{T}} . So even though a symmetric tensor has a symmetric matrix ($A^T = A$) and a rotation matrix is orthogonal ($A^{-1} = A^T$), these properties are unrelated to each other. The tangential component is directed oppositely to that on the right-hand face, but the torque it exerts is the same. Now, consider the force acting at point B, which is located the same distance below the $x_2$-axis. Find the first three non-zero terms of the Taylor series of f. Delete the space below the header in moderncv. The best answers are voted up and rise to the top, Not the answer you're looking for? Then $C$ is definitely not the same as $BA$. The best answers are voted up and rise to the top, Not the answer you're looking for? Is there precedent for Supreme Court justices recusing themselves from cases when they have strong ties to groups with strong opinions on the case? The point is that the tensor holds regardless of the change in coordinates. Consider the point labelled A, which is located above the $x_2$-axis at $x_2$ = $\Delta/2$. My advisor refuses to write me a recommendation for my PhD application unless I apply to his lab. \bar{n}^{(1)} \cdot A \cdot n ^{(1)}=\lambda_{1} \bar{n}^{(1)^{T}} n ^{(1)}, \bar{n}^{(1)^{T}} \cdot A \cdot n ^{(1)}=\overline{\lambda_{1}} \bar{n}^{(1)^{T}} n ^{(1)}, \begin{array}{l}\bar{n}^{(1)^{T}} \cdot A \cdot n ^{(1)}-\bar{n}^{(1)^{T}} \cdot A \cdot n ^{(1)}=\overline{\lambda_{1}} \bar{n}^{(1)^{T}} n ^{(1)}-\lambda_{1} \bar{n}^{(1)^{T}} n ^{(1)} \text {, }\\0=\left(\bar{\lambda}_{1}-\lambda_{1}\right) \bar{n}^{(1)^{T}} n ^{(1)} \text {. (When is a debt "realized"?). If it doesn't work, it's not that you don't have a tensor. I'm sure General Tensors would have any jacobian and inverse jacobians are matrices rather than just the Lorentz transformations. Your email address will not be published. Like $T_{ab} (v_a \frac{\partial}{\partial x^a})(v_b \frac{\partial}{\partial x^b})$? Prove that a real symmetric tensor, A, has real eigenvalues. jk is symmetric. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. \bar{n}^{(1)^{T}} \cdot A =\bar{\lambda}_{1} \bar{n}^{(1)^{T}} . -----l3v1-----recommendations 12:35 i should took the other half, because the fulcrum is on the other side, later way at the back will explain, s. The torque per unit area at any point is the stress vector crossed with the moment arm $\vec{r}$ (Figure $\PageIndex{2}$b), which is the perpendicular distance from the 1 axis to the point where the force acts. Dear Jayadeep: "1. Where \Lambda are the Lorentz transformation matrices (translations, rotations, or boosts). rev2022.12.7.43084. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. What is the best method for showing that a tensor is a symmetric tensor? 22.1 Tensors Products We begin by dening tensor products of vector spaces over a eld and then we investigate some basic properties of these tensors, in particular the existence of bases and duality. Now expand $\tau_{23}$ in a first-order Taylor series about the origin: \[\tau_{23}\left(x, \frac{\Delta}{2}, z\right)=\tau_{23}^{0}+\frac{\partial \tau_{23}^{0}}{\partial x} x+\frac{\partial \tau_{23}^{0}}{\partial y} \frac{\Delta}{2}+\frac{\partial \tau_{23}^{0}}{\partial z} z+O\left(\Delta^{2}\right).\label{eqn:1} \], The superscript 0 denotes the value of $\tau_{23}$ or one of its derivatives evaluated at the origin. Decomposition of a Cartesian tensor. Do mRNA Vaccines tend to work only for a short period of time? Is playing an illegal Wild Draw 4 considered cheating or a bluff? Ultimately, the only difference is that the stress is evaluated at $y$ = $-\Delta/2$ rather than $y$ = $\Delta/2$, so that, \[T_{1}^{[l e f t]}=\Delta^{2} \frac{\Delta}{2}\left(\tau_{23}^{0}-\frac{\partial \tau_{23}^{0}}{\partial y} \frac{\Delta}{2}\right). You can find the components of the velocity in your system of coordinates: $u_x^i(t) = dx^i/dt$. There's no contradiction. What is the recommender address and his/her title or position in graduate applications? This is not an important distinction here. Note that the terms in Equation $\ref{eqn:1}$ proportional to $x$ and $z$ have integrated to zero. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. To learn more, see our tips on writing great answers. I still do not know how to show that a symmetric tensor is indeed a tensor? I also understand that the permutation equals zero if the tensor $T_{ij}$ is symmetric because that would mean that i = j. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Can you end a sentence with respectively. What if my professor writes me a negative LOR, in order to keep me working with him? rev2022.12.7.43084. Therefore to show something is a tensor you just have to show that it obeys the transformation equation and that your transformed answer is still a valid result and can be transformed back to the original by doing the inverse transform. If the result is the same for general $a, b$ then you have your proof. $$II_{ab}=\nabla^T_a \hat{n}_b \big\vert_\Sigma$$ Also if you want to give a student like myself who is new to tensor some advice on learning tensors, and some tensor properties, and how to work with them, be my guest :), Also, are all transformations homogeneous linear transformations? First, suppose you have some space (it can be 3-space or spacetime or whatever) and you have a set of coordinates $\{x^i\}$ defined on it. The point is that the tensor holds regardless of the change in coordinates. why i see more than ip for my site when i ping it from cmd. Therefore, it makes sense to say that symmetry is a property of the tensor instead of its representation in a particular coordinate system. "BUT" , sound diffracts more than light. Prove that every bilinear map can be written as a sum of bilinear symmetric map and a bilinear anti-symmetric map. Moreover this map is linear by construction. For special relativity this is the Lorentz transform, but in classical physics in may be a simple rotation. If it's for boosts then we have a matrix with hyperbolic functions (I cannot remember if this is orthogonal as well?). Do inheritances break Piketty's r>g model's conclusions? stress tensor), but is a more general concept that describes the linear relationships between objects, independent of the choice of coordinate system. I have used the chain rule and the fact that the $y^i$ are functions of the $x^j$. How would we prove this is a tensor? Why did NASA need to observationally confirm whether DART successfully redirected Dimorphos? As drawn here, $\tau_{32}$ > 0, and the torque is clockwise. In general, it's a bad idea to trawl for online sources because conventions will differ. Therefore to show something is a tensor you just have to show that it obeys the transformation equation and that your transformed answer is still a valid result and can be transformed back to the original by doing the inverse transform. I think this may help, thinking of these tensors as matrices themselves, visually. }. Use MathJax to format equations. (Depending on the source, this might also be called the Levi-Civita symbol. Is my question about proving whether or not $S^{\mu\nu}$ equivalent to showing that a 2nd order symmetric tensor remains symmetric when transformed into any other coordinate system? If the result is the same for general a, b then you have your proof. We'll do this in two ways: the rst is intuitive and physically transparent, and the second is a bit technical and uses the machinery of continuum theories. Asking for help, clarification, or responding to other answers. I have used the chain rule and the fact that the $y^i$ are functions of the $x^j$. 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Prove that a symmetric matrix P is positive definite iff all eigenvalues of P are positive. A tensor is not particularly a concept related to relativity (see e.g. @MycrofD say we have matrices $A$, $B$ with $C = AB$. Well, you're not the first. Connect and share knowledge within a single location that is structured and easy to search. I even found a question saying that in general they are non-commutative. On the bottom face, the force is again opposite but the torque is the same. A tensor is not particularly a concept related to relativity (see e.g. Connect and share knowledge within a single location that is structured and easy to search. Assume that the stress tensor is uniform in space. Also we can think of $\Lambda$ if it's a rotation matrix having the property $\Lambda^{-1}=\Lambda$. Taking the transpose of Equation (1.17) and making use of these two properties gives Moreover this map is linear by construction. This coordinate independence results in the transformation law you give where, \Lambda, is just the transformation between the coordinates that you are doing. Also we can think of $\Lambda$ if it's a rotation matrix having the property $\Lambda^{-1}=\Lambda$. Nothing whatsoever needs to be checked. Our professor defined a rank $(k,l)$ tensor as something that transforms like a tensor. Can you please show some reference? Prove that a symmetric matrix P is positive definite iff all eigenvalues of P are positive. The link doesn't go any farther than that. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. tensors Share Cite Follow Thanks for clarifying. What's the benefit of grass versus hardened runways? $\Lambda^i_{\ j} = \frac{\partial y^i}{\partial x^j}$. A tensor is a coordinate independent object, and its matrix will change if you change coordinates. The magnitude of that cross product is just $|\tau_{23}|$ times $|\vec{r}| \cos \theta$, where $\theta$ is the angle between $\vec{r}$ and the horizontal, and this in turn is equal to $|\tau_{23}|\Delta/2$. What should my green goo target to disable electrical infrastructure but allow smaller scale electronics? (1.19), Multiplying Equation (1.18) by n^{(1)} on the right gives, \bar{n}^{(1)^{T}} \cdot A \cdot n ^{(1)}=\overline{\lambda_{1}} \bar{n}^{(1)^{T}} n ^{(1)}. China 2013, 8(1): 19-40 DOI 10.1007/s11464-012-0262-x Best rank one approximation of real symmetric tensors can be chosen symmetric Shmuel FRIEDLAND Departmentof Ma Now we can define vectors in general, by asking that they have the same transformation law as velocities: A vector $\vec{X}$ is a function that assigns a set of numbers (called its components) $X_x^i\ (i = 1\dots n)$ to each coordinate system $\{x^i\}$, such that if $\{x^i\}$ and $\{y^i\}$ are two coordinate systems, the components of $X$ are related by, $$X_y^i = \frac{\partial y^i}{\partial x^j} X_x^j$$. Another way to show this is to explicitly compute how the components change (as Carroll did in (2.67)). Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. (When is a debt "realized"? The force acting there is $\tau_{3j}\hat{e}^{(j)}$, and its tangential component is $\tau_{32}\hat{e}^{(2)}$. What factors led to Disney retconning Star Wars Legends in favor of the new Disney Canon? As an exercise, write $BA$ in component notation and see the difference. First, suppose you have some space (it can be 3-space or spacetime or whatever) and you have a set of coordinates $\{x^i\}$ defined on it. Right! Physically, the only thing that the electromagnetic field tensor and a Lorentz transformation generator have in common is that they both happen to be antisymmetric rank 2 tensors. Symmetric tensors were studied in details in and , where the authors also work with a Jacobi-type algorithm. Did they forget to add the layout to the USB keyboard standard? A tensor is nothing more or less than a linear map from (possibly multiple copies of) a vector space (and possibly copies of its dual space) into the scalar field. Symmetry may not even be meaningful for, e.g. How could a really intelligent species be stopped from developing? }\end{array} (1.21). But I think when I read the problems I interpret it to asks for the general case of $\Lambda^{\mu}$, but I do not know. The symmetry of the stress tensor will be demonstrated in two ways. How to characterize the regularity of a polygon? Help us identify new roles for community members, Abstract formulation of the Riemann-tensor in index notation, Isotropic tensor functions that map antisymmetric tensors to zero (Navier-Stokes derivation). It is often helpful to regard such a vector as an object $\vec{u}$ that is independent of coordinates. Aligning vectors of different height at bottom. Therefore, it makes sense to say that symmetry is a property of the tensor instead of its representation in a particular coordinate system. You ask how to prove that a symmetric tensor is a tensor, but this is a tautological question, because a symmetric tensor obviously is a tensor! $$ 0 = \epsilon_{klm} \epsilon_{ijk} T_{ij} = (\delta_{il} \delta_{jm} - \delta_{im} \delta_{jl}) T_{ij} = T_{lm} - T_{ml} , $$ But this is a purely mathematical analogy. I hope that you'll learn to appreciate both. \nonumber \], The net torque on the right and left faces is, \[T_{1}^{[r i g h t]}+T_{1}^{[l e f t]}=\Delta^{3} \tau_{23}^{0}. Now let's get to your question. However, this coincidence does lead to a few analogies. We now conclude that the net torque about $\hat{e}^{(1)}$ is proportional to the difference between $\tau_{23}$ and $\tau_{32}$. More precisely, a metric tensor at a point p of M is a bilinear form defined on the tangent space at p (that is . Also we can think of $\Lambda$ if it's a rotation matrix having the property $\Lambda^{-1}=\Lambda$. Also if you want to give a student like myself who is new to tensor some advice on learning tensors, and some tensor properties, and how to work with them, be my guest :), Also, are all transformations homogeneous linear transformations? The general rule with the antisymmetric tensor: if in doubt, multiply by another one and use the $\epsilon \epsilon = \delta\delta-\delta\delta$ identity. A tensor $T_{ab}$ of rank $2$ is symmetric if, and only if, $T_{ab}=T_{ba}$, and antisymmetric if, and only if, $T_{ab}=-T_{ba}$. Why is it so hard to convince professors to write recommendation letters for me? Thanks for clarifying. \left.\left.\overline{\left(\overline{A \cdot n^{(1)}}\right. What was the last x86 processor that didn't have a microcode layer? Do sandcastles kill more people than sharks? Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. This lecture gives a nice matrix form of what a symmetric (2,0) tensor looks like. This coordinate independence results in the transformation law you give where, $\Lambda$, is just the transformation between the coordinates that you are doing. Then Can we think of the EM tensor as an infinitesimal generator of Lorentz transformations? Then you can also apply the transformation to check. After this, we investigate special kinds of tensors, namely, symmetric tensors and skew-symmetric tensors. Then Equation (1.21) requires that 0=\left(\bar{\lambda}_{1}-\lambda_{1}\right) , which says that the eigenvalue is equal to its complex conjugate. A transformation is defined only between a specific pair of coordinate systems. So let the second fundamental form act on two vectors, then interchange the indices and let it act on the same two vectors? We prove that the algorithm converges to a stationary point of the objective function. Our professor defined a rank $(k,l)$ tensor as something that transforms like a tensor as follows: $$T^{\mu_1' \mu_2'\mu_k'}{}_{\nu_1'\nu_2'\nu_l'} ~=~ \Lambda^{\mu_1'}{}_{\mu_1}\Lambda^{\mu_k'}{}_{\mu_k}~\Lambda^{\nu_1}{}_{\nu_1'}\Lambda^{\nu_l}{}_{\nu_l'}~T^{\mu_1\mu_2\mu_k}{}_{\nu_1\nu_2\nu_l}$$. I suspect that the actual question is as follows. Remember that there's an implied summation sign. So now we know how the velocity of a particle (or, as the mathematicians would call it, the tangent vector to a curve) transforms when you change coordinates. The first is fairly intuitive. Will a Pokemon in an out of state gym come back? Why are Linux kernel packages priority set to optional? The Levi-Civita symbol is the one whose components are all $0$ or $\pm 1$, and contraction with it yields the orientation of an ordered set of basis vectors. Prove that a symmetric matrix P is positive definite iff all eigenvalues of P are positive. All too common, and all too pedagogically faulty. But lets just talk about a rank (2,0) symmetric contravariant tensor for a second, denoted SS^{\mu\nu} and equals SS^{\nu\mu}. You defined a symmetric tensor as one that has the property $T^{ij} = T^{ji}$. Next we consider the upper face. Can LEGO City Powered Up trains be automated? The components of a vector (or a tensor) will depend on the coordinates, but if everything transforms the same way, equations made out of tensors will have the same form in different coordinate systems. So, in your case, if you let the second fundamental form act on two basis vectors $e_i$ and $e_j$, or you interchange them. I'm not sure if this is only for SR or if also for GR since we've only been talking about SR thus far, though GR is something we'll be covering soon. Transcribed Image Text: 5. Example with the metric tensor we have distance between 2 points as being invariant under coordinate transformation. Look at how the tensor acts on the two vectors $a$ and $b$. Save my name, email, and website in this browser for the next time I comment. Here $t$ is just a parameter. Therefore, \overline{ n }^{(1)^{T}} n ^{(1)} \neq 0. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The Levi-Civita symbol doesn't scale with volume, which is why its transformation has an extra power of the determinant. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. It only takes a minute to sign up. Why are Linux kernel packages priority set to optional? So it might be a good idea to check that this holds. If $\tau_{23}$ is positive as shown, then the torque is positive (i.e., counterclockwise). Counting distinct values per polygon in QGIS. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. The tensor A is real, which means the tensor is equal to its complex conjugate, A =\bar{A}, and it is symmetric, A = A ^{T}. Why do we order our adjectives in certain ways: "big, blue house" rather than "blue, big house"? (I am using subscripts to label coordinate systems.) With the help of the notation of the skew-symmetric tensor, the expression for the 1st order absolute and relative derivatives of a tensor was presented, and the expression for the 2nd order absolute and relative derivatives of a tensor was offered further. If the tensor is symmetric, then, those torques add up to zero. Does this picture below do anything for the problem? You're saying though that this equals something in which I lost no information & then it's invertible with no loss of information? Thanks for contributing an answer to Mathematics Stack Exchange! So we have found out that if a tensor is symmetric in some coordinate system, it is symmetric in any coordinate system. Otherwise what could I do with an expression such as $\Lambda^{\mu}\Lambda^{\nu}T^{\mu\nu}$? And then compare this to the result when you interchanged $a$ and $b$. Challenges of a small company working with an external dev team from another country. How would we prove this is a tensor? General Relativity - Curvature of two superimposed perfect pressureless fluids. I even got the implied summation you meant, which was initially not obvious to me. Explicitly, we have for the entropy, S S0 = (1=2)jkPjPk, where S0 is the entropy at thermodynamic equilibrium, so that the prob- ability distribution function for the Pj may be written w(Pj) = p det (2)3=2 exp 1 2 jkPjPk: (3) To obtain, Eq. http://www.lecture-notes.co.uk/susskind/special-relativity/lecture-6/rank-two-tensors/ How to characterize the regularity of a polygon? This is question in Prof. Zee's, "Einstein's Gravity in a Nutshell", Chapter I.4 Exercises 2. Why didn't Democrats legalize marijuana federally when they controlled Congress? Reason for asking: Under what conditions would a cybercommunist nation form? I have been searching the net, but could find no reliable source. Then $C$ is definitely not the same as $BA$. Deriving the Epsilon-Tensor (Levi-Civita Symbol), Levi-Civita tensor contraction contradiction, Cofactor expression with Levi Civita symbol. What is the coordinate-free definition of the Levi-Civita Symbol? It may not be in my best interest to ask a professor I have done research with for recommendation letters. Solution 1. Then you can also apply the transformation to check. So from this definition you can easily check that this decomposition indeed yields a symmetric and antisymmetric part. It only takes a minute to sign up. So what symmetry in fact means is that $T_{ij}= T_{ji}$. Where was Wolverine during X-Men: Apocalypse? How could a really intelligent species be stopped from developing? Here, there are two different concepts: the Levi-Civita symbol, and the Levi-Civita tensor. Also if you want to give a student like myself who is new to tensor some advice on learning tensors, and some tensor properties, and how to work with them, be my guest , Also, are all transformations homogeneous linear transformations? What is the relationship between AC frequency, volts, amps and watts? $\partial y^i / \partial x^j$ will have different properties depending on the coordinates. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. But if you know the coordinates $y^i$ as a function of the coordinates $x^j$, you can find out how the two velocities are related: $$u_y^i(t) = \frac{dy^i(x)}{dt} = \frac{\partial y^i}{\partial x^j} \frac{d x^j}{dt} = \frac{\partial y^i}{\partial x^j} u^j_x(t)$$. $T$ on the left is a single number equal to the sum of all $\Lambda\Lambda T$ terms on the right (for certain i,j,y). Also we can think of \Lambda if its a rotation matrix having the property 1=\Lambda^{-1}=\Lambda. Then if somebody tells you the electromagnetic field is the same kind of tensor, you'll automatically know that it can be broken down into two three-vectors, namely the electric and magnetic fields. rev2022.12.7.43084. To get the second equality I used that $T_x^{kl} = T_x^{lk}$, to get the third equality I moved the $\Lambda$s around, and in the first and last equalities I used the transformation law for a tensor. If I understand correctly, you're asking how to prove that symmetry of a tensor is coordinate independent, but you seem to be having trouble with the definition of a tensor. But in component notation, $C_{ij} = \sum_k A_{ik} B_{kj}$, and this is the same as $\sum_k B_{kj} A_{ik}$, because the components are just numbers; the sum is a sum of products of numbers. I'm sure General Tensors would have any jacobian and inverse jacobians are matrices rather than just the Lorentz transformations. Giving examples of some group $G$ and elements $g,h \in G$ where $(gh)^{n}\neq g^{n} h^{n}$. is it not possible to show that for any $A_{bc}$, $1/2(A_{bc} + A_{cb}$ is symmetric?that's what I want to see. Edited the answer for clearer explanation. I'm not sure if this is only for SR or if also for GR since we've only been talking about SR thus far, though GR is something we'll be covering soon. Alternatively, a symmetric tensor of order r represented in coordinates as a quantity with r indices satisfies To learn more, see our tips on writing great answers. When does money become money? Simply consider the vector components in the same coordinate system, and contract indices: $T(\vec{v},\vec{w}) = T_{\mu\nu} v^\mu w^\nu \in \mathbb{R}$. coordinate systemsgeneral-relativityspecial-relativitysymmetrytensor-calculus. Click hereto get an answer to your question If R and S are relations on a set A , then prove the following:(i) R and S are symmetric R S and R S are symmetric(ii) R is reflexive and S is any relation R S is reflexive. How would we prove this is a tensor? Front. Help with tensor calculus identity proof (antisymmetric matrix and levi civita symbol) 1. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); http://www.math.ucla.edu/~baker/149.1.02w/handouts/e_htls.pdf, http://www.lecture-notes.co.uk/susskind/special-relativity/lecture-6/rank-two-tensors/, Schwarzschild metric in lower-dimensional spaces. The only independent components are the, The reader should take note that the specific duality we have just described is unique to three-dimensional space; in four dimensions (appropriate for relativity) an antisymmetric rank-2 tensor has, In mathematics, the conjugate transpose (or Hermitian transpose) of an m-by-n matrix with complex entries is, In differential geometry, the components of a vector relative to a basis of the tangent bundle are. $$ \epsilon_{ijk} \epsilon_{klm} = \delta_{il} \delta_{jm} - \delta_{im} \delta_{jl} $$ Also, I am restricting myself to coordinate bases for simplicity. Our Website is free to use.To help us grow, you can support our team with a Small Tip. More specifically, Basically theyre symmetric matrices of the form AT=AA^T=A. I think this may help, thinking of these tensors as matrices themselves, visually. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. I get what you are saying, somewhat. But let's just talk about a rank (2,0) symmetric contravariant tensor for a second, denoted $S^{\mu\nu}$ and equals $S^{\nu\mu}$. The book I have for tensor calculus is very introductory and does not go in depth into the basics of Einstein notation. The first is fairly intuitive. Is there hope for Einstein tensor notation in Quantum Mechanics? If I give you components $T_{\mu\nu}$ (16 components in all) in a specific coordinate system/basis, then you can map any two vectors into a scalar with it. why i see more than ip for my site when i ping it from cmd. I'm completely lost on this. In order to have a symmetry-preserving Jacobi-type algorithm, rotation matrices should be the same in . As stated in the link you gave, this result doesn't allow us to think of electromagnetism as a geometric phenomenon, because different particles have different values of, Be careful to distinguish between active and passive Lorentz transformations. A tensor is just an abstract quantity that obeys the coordinate transformation law. Oh dear. In general, sigma_ij is a function of coordinates x, y and z (i.e., it can vary from a position to another). How to prove a symmetric tensor is indeed a tensor? He wrote on the whiteboard: if S=SS_{\mu\nu\rho}=S_{\nu\mu\rho} then SS is symmetric in \mu and \nu. Six independent components of the stress tensor. \nonumber \], Now the rotational form of Newtons second law states that this torque equals $I_{11}\alpha_1$, where $I_{11}$ is the moment of inertia for torque and rotation about $\hat{e}^{(1)}$ and $\alpha_1$ is the corresponding angular rotation. If you have a matrix $\Lambda^i_{\ j}$ relating coordinates $x$ and $y$ as above, it makes no sense to ask what $\Lambda$ looks like in coordiantes $z$. I think this may help, thinking of these tensors as matrices themselves, visually. Why did NASA need to observationally confirm whether DART successfully redirected Dimorphos? Is there a proof, or is this just a definition? @MycrofD when using component notation, everything is numbers, so it doesn't matter what order you write the factors in. 0. Basically they're symmetric matrices of the form $A^T=A$. A tensor is a coordinate independent object, and its matrix will change if you change coordinates. Then, $$T_y^{ij} = \Lambda^i_{\ k} \Lambda^j_{\ l} T^{kl}_x = \Lambda^i_{\ k} \Lambda^j_{\ l} T^{lk}_x = \Lambda^j_{\ l} \Lambda^i_{\ k} T^{lk}_x = T_y^{ji}$$. a two-point tensor (a multilinear map between two separate manifolds) like deformation gradient in continuum mechanics. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. This lecture gives a nice matrix form of what a symmetric (2,0) tensor looks like. The components of a vector (or a tensor) will depend on the coordinates, but if everything transforms the same way, equations made out of tensors will have the same form in different coordinate systems. Following the definition of a vector, we can define a $(2,0)$ tensor $T$ (not necessarily a product of vectors as above) as a function that assigns a set of numbers $T^{ij}_x$ to each coordinate system, such that the components in two different systems follow the above transformation law. This is question in Prof. Zee's, "Einstein's Gravity in a Nutshell", Chapter I.4 Exercises 2. I know that ij are the dummy indices and that k is a free index. So if I'm proving that it's a tensor in general, I will need to know all of the properties of $\Lambda$? If the tensor is symmetric, then, those torques add up to zero. If we apply an electric field, the three-velocity grows in the direction of the field, just like it does in the direction of a boost. Transformation of the Levi Civita symbol - Carroll, Question about Wald's example of a "derivative operator", Proof that a Lorentz-invariant scalar function can only depend on scalar products. Edit: Let $S_{bc}=\dfrac{1}{2}\left(A_{bc}+A_{cb}\right)$. Rather, you have two distinct tensors. In General Relativity we use all kinds of coordinates, and the transformations will not in general be linear. So we have found out that if a tensor is symmetric in some coordinate system, it is symmetric in any coordinate system. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. http://www.lecture-notes.co.uk/susskind/special-relativity/lecture-6/rank-two-tensors/ For example, you have expressions for all 16 $T_{\mu\nu}$ and another 16 $T_{\mu'\nu'}$. Well, you're not the first. To learn more, see our tips on writing great answers. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. So we have found out that if a tensor is symmetric in some coordinate system, it is symmetric in any coordinate system. Changing the style of a line that connects two nodes in tikz, When does money become money? May 12, 2022 by grindadmin. . What am I missing? stress tensor), but is a more general concept that describes the linear relationships between objects, independent of the choice of coordinate system. For special relativity this is the Lorentz transform, but in classical physics in may be a simple rotation. On the left-hand face, the applied force is reversed because the unit normal is $-\hat{e}^{(2)}$. This coordinate independence results in the transformation law you give where, $\Lambda$, is just the transformation between the coordinates that you are doing. What am I missing? Im not sure if this is only for SR or if also for GR since weve only been talking about SR thus far, though GR is something well be covering soon. So it might be a good idea to check that this holds. Connect and share knowledge within a single location that is structured and easy to search. Let $\{y^i\}$ be an arbitrary coordinate system. Any tensor of rank 2 can be rewritten as: $$A_{bc} = \frac{1}{2}(A_{bc} + A_{cb}) + \frac{1}{2}(A_{bc}-A_{cb})$$. (1.17), The tensor A is real, which means the tensor is equal to its complex conjugate, A =\bar{A}, and it is symmetric, A = A ^{T}. Was this reference in Starship Troopers a real one? If we start with a nonzero three-velocity and apply a magnetic field, the velocity spins around. An example is. Prove a tensor is symmetric using Levi-Civita symbol Asked 3 years, 3 months ago Modified 3 years, 3 months ago Viewed 601 times 0 Specific Question if i j k T i j = 0 show T i j = T j i Reason for asking: The book I have for tensor calculus is very introductory and does not go in depth into the basics of Einstein notation. CGAC2022 Day 6: Shuffles with specific "magic number". - Grego_gc Feb 6, 2020 at 15:05 @Grego_gc I think the OP is saying 2 y x = y gives equilibrium in y direction. And let's say you have a particle moving in your space, with a trajectory given by $x^i = x^i(t)$. Corollary 1. A tensor is not particularly a concept related to relativity (see e.g. I can understand how that works. If it's for boosts then we have a matrix with hyperbolic functions (I cannot remember if this is orthogonal as well?). Step 1: First, check if it's a square matrix, as only square matrices can be considered as symmetric matrices. How would we prove this is a tensor? How did Chandragupta II actions help the Gupta empire? Let me give you a definition that might help. But it's just a sanity check. I'm sure General Tensors would have any jacobian and inverse jacobians are matrices rather than just the Lorentz transformations. }, \left.\left.\overline{\left(\overline{A \cdot n^{(1)}}\right. \end{aligned} \nonumber \]. For special relativity this is the Lorentz transform, but in classical physics in may be a simple rotation. Why is Julia in Cyrillic regularly transcribed as Yulia in English? sr] (mathematics), Symmetric part of any matrix A is given by, The first example to look at is a tensor with two indices Tab. Under any coordinate transformation that preserves the right-handedness of the coordinate system, the components of the Levi-Civita symbol stay the same. The derivatives of the transformation $\partial y^i / \partial x^j$ can also be represented as a matrix. Rather, you have two distinct tensors. Consider a cube with edge length $\Delta$, as shown in Figure $\PageIndex{1}$a, and the distribution of forces that act to rotate the cube counterclockwise about $\hat{e}^{(1)}$ (blue arrow). He wrote on the whiteboard: if $S_{\mu\nu\rho}=S_{\nu\mu\rho}$ then $S$ is symmetric in $\mu$ and $\nu$. Why is it natural to make the electromagnetic field an antisymmetric, rank 2 tensor? The point is that the tensor holds regardless of the change in coordinates. This is not an important distinction here. Proof that terms in decomposition of a tensor are symmetric and antisymmetric, Help us identify new roles for community members, Show that the symmetry properties of a tensor are invariant, Decomposition of the symmetric part of a tensor, Help with tensor calculus identity proof (antisymmetric matrix and levi civita symbol), Deriving covariant derivative identitiy of an antisymmetric tensor. I might have expressed myself badly. Why don't courts punish time-wasting tactics? Therefore, it makes sense to say that symmetry is a property of the tensor instead of its representation in a particular coordinate system. $$\frac{d u_\mu}{d\tau} = (q/m) F_{\mu\nu} u^\nu.$$ So let's suppose in some coordinates $\{x^i\}$ it happens that $T_x^{ij} = T_x^{ji}$ for all $i,j$. Separating columns of layer and exporting set of columns in a new QGIS layer. Counting distinct values per polygon in QGIS, why i see more than ip for my site when i ping it from cmd. Does an Antimagic Field suppress the ability score increases granted by the Manual or Tome magic items? By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. A tensor is not particularly a concept related to relativity (see e.g. So if you define it there, then there are many different ways to extend the definition to general bases: Because every source will have different conventions on nomenclature, Wikipedia (which itself is compiled inconsistently from many sources) is not reliable. How to prove a symmetric tensor is indeed a tensor? Can you please show some reference? Symmetric Tensor Theorem | Tensor Algebra. Where $\Lambda$ are the Lorentz transformation matrices (translations, rotations, or boosts). where $u^\mu$ is the four-velocity; you can expand this in components to verify it's just the Lorentz force law. But I think when I read the problems I interpret it to asks for the general case of $\Lambda^{\mu}$, but I do not know. $$\frac{1}{2}(A_{bc}-A_{cb})$$ is antisymmetric. So even though a symmetric tensor has a symmetric matrix ($A^T = A$) and a rotation matrix is orthogonal ($A^{-1} = A^T$), these properties are unrelated to each other. http://www.lecture-notes.co.uk/susskind/special-relativity/lecture-6/rank-two-tensors/ Why didn't Democrats legalize marijuana federally when they controlled Congress? What's the translation of "record-tying" in French? For a symmetric tensor A, there exist (at least) three eigenvectors that are mutually orthogonal. Therefore to show something is a tensor you just have to show that it obeys the transformation equation and that your transformed answer is still a valid result and can be transformed back to the original by doing the inverse transform. 1 Answer Sorted by: 1 Look at how the tensor acts on the two vectors a and b. Why is it important that Hamiltons equations have the four symplectic properties and what do they mean? Why "stepped off the train" instead of "stepped off a train"? But that would be possible only if lorentz transformation is commutative. \nonumber \], The torques on the top and bottom faces are calculated in the same manner, and give $-\Delta^3\tau^0_{32}$, so that the net torque about $\hat{e}^{(1)}$ is, \[T_{1}=\Delta^{3}\left(\tau_{23}^{0}-\tau_{32}^{0}\right). 5. - Feb 6, 2020 at 15:15 He wrote on the whiteboard: if $S_{\mu\nu\rho}=S_{\nu\mu\rho}$ then $S$ is symmetric in $\mu$ and $\nu$. Is $X^{\mu\nu} \equiv A^{\mu}+B^{\nu}$ a tensor? Proof Since A is a symmetric tensor, it has got exactly three eigenvalues, 1, 2, 3, by Theorem 2.13.2. Our algorithm is not structure-preserving. Can I cover an outlet with printed plates? The best answers are voted up and rise to the top, Not the answer you're looking for? Remember that there's an implied summation sign. the symmetric n-fbld tensor products endowed with the projective topology. The symmetry of the stress tensor will be demonstrated in two ways. This is essentially the same as the "set of numbers that transforms like this" definition, but I find it to be a bit clearer and more explicit as to what things are. and $T_{bc}$ is antisymmetric. Im sure General Tensors would have any jacobian and inverse jacobians are matrices rather than just the Lorentz transformations. This is question in Prof. Zee's, "Einstein's Gravity in a Nutshell", Chapter I.4 Exercises 2. Right! Did they forget to add the layout to the USB keyboard standard? How to calculate pick a ball Probability for Two bags? Also if you want to give a student like myself who is new to tensor some advice on learning tensors, and some tensor properties, and how to work with them, be my guest , Also, are all transformations homogeneous linear transformations? A tensor can be defined as something that transforms as products of vectors: If we take two vectors $\vec{u}$ and $\vec{v}$ and define the (coordinate-dependent) quantity $T_x^{ij} = u^i_x v^j_x$, then in two different coordinate systems we find (defining $\Lambda^i_{\ j} = \frac{\partial y^i}{\partial x^j}$): $$T^{ij}_y = \Lambda^i_{\ k} \Lambda^j_{\ l} T^{kl}_x$$. the components of a symmetric d d matrix. Why "stepped off the train" instead of "stepped off a train"? The stress tensor is symmetric if the body is in equilibrium. I get what you are saying, somewhat. Does this picture below do anything for the problem? If $\tau_{23}$ = $\tau_{32}$, the net torque about $\hat{e}^{(1)}$ is zero. Our professor defined a rank $(k,l)$ tensor as something that transforms like a tensor. On the right-hand face, the unit normal is $\hat{e}^{(2)}$. (1.16), Taking the complex conjugate of this equation gives, \overline{A \cdot n^{(1)}}=\overline{\lambda_{1} n^{(1)}} \text {. Math. You defined a symmetric tensor as one that has the property $T^{ij} = T^{ji}$. But let's just talk about a rank (2,0) symmetric contravariant tensor for a second, denoted $S^{\mu\nu}$ and equals $S^{\nu\mu}$. Does the result match the definition of (anti-)symmetry? Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. You're saying though that this equals something in which I lost no information & then it's invertible with no loss of information? Integrating $\tau_{23}\Delta/2$ over the right-hand face, we find, \[\begin{aligned} We will address these issues in the more rigorous version that follows. We would like to prove that if the above identity is true in one coordinate system, it is true in all of them. $$\frac{1}{2}(A_{bc} + A_{cb})$$ is symmetric, and Oh dear. Let v1, v2 and v3 be the corresponding eigenvectors. (the way to remember this: cycle the indices on the $\epsilon$s so that the first letter is the same, then the $+$ is on the terms with the indices paired in the same order, the $-$ is on the terms in the opposite order). Continuum Mechanics natural to make the electromagnetic field an antisymmetric, rank 2 tensor me though. N'T want to see how these terms being symmetric and antisymmetric part vector as an exercise, write BA! A negative LOR, in order to keep me working with an dev... J } = T_ { ji } $ using component notation and the! Delete the space below the main diagonal represent torques that are mutually orthogonal infinitesimal generator Lorentz., not the answer you 're looking prove a tensor is symmetric in coordinates the objective function to disable infrastructure. N'T go any farther than that a bad idea to check sum of bilinear symmetric and! N^ prove a tensor is symmetric ( 1 ) } } \right free index superimposed perfect pressureless fluids tensor on... $ BA $ layer and exporting set of columns in a particular coordinate system, it invertible! Picture below do anything for the next time i comment Lorentz transformation matrices ( translations, rotations or. As Carroll did in ( 2.67 ) ) break Piketty 's r > g model 's conclusions from cases they. The top, not the answer you 're saying though that this holds find reliable. Face, the velocity spins around 're looking for book i have for tensor calculus very... To groups with strong opinions on the bottom face, the unit normal is \ ( {. Symbol does n't scale with volume, which is why its transformation has an power. Clarification, or boosts ) see how these terms being symmetric and antisymmetric part writing a reference not. Design / logo 2022 Stack Exchange is a debt `` realized ''? ), volts amps... '', Chapter I.4 Exercises 2 a recommendation for my site when i it. \Partial y^i / \partial x^j $ will have different properties Depending on the source, coincidence... To characterize the regularity of a line that connects two nodes in tikz, when does become... A free index Sorted by: 1 look at how the tensor acts on the.. Then the torque is clockwise is in equilibrium successfully redirected Dimorphos notation, is. ) -axis recommendation for my site when i ping it from cmd help! Few analogies there are two different concepts: the Levi-Civita tensor is indeed a tensor to verify 's. Any level and professionals in related fields an antisymmetric, rank 2 tensor looking for me! Exist ( at least ) three eigenvectors that are equal but opposite website is free to use.To help grow! Period of time record-tying '' in French magic number '' if we start with a nonzero three-velocity and apply magnetic! \Overline { a \cdot n^ { ( 2 ) } \ ) > 0, website... And share knowledge within a single location that is structured and easy to search to see how these being... Match the definition of ( anti- ) symmetry no reliable source the dummy indices and it... Symmetry may not be in my best interest to ask a professor i have for tensor calculus is very and... Hope for Einstein tensor notation in Quantum Mechanics an arbitrary coordinate system, it a! ) three eigenvectors that are equal but opposite you 're looking for the unit normal is \ ( {... Of layer and exporting set of columns in a Nutshell '', sound diffracts more than ip for site! Positive ( i.e., counterclockwise ) 1.17 ) and making use of these two properties gives Moreover this is. And his/her title or position in graduate applications result when you interchanged $ a $, b... Tensor is not particularly a concept related to relativity ( see e.g $ T_ { }! Multilinear map between two separate manifolds ) like deformation gradient in continuum Mechanics next time i comment that. The next time i comment the electromagnetic field an antisymmetric, rank 2?... The definition of the Taylor series of f. Delete the space below the header in moderncv the layout to result... Extra power of the transformation $ \partial y^i / \partial x^j $ can apply... Tikz, when does money become money assume that the algorithm converges to stationary. With him a microcode layer same for general a, there exist ( at least ) eigenvectors. ( anti- ) symmetry these tensors as matrices themselves, visually nodes in tikz, when money... Change ( as Carroll did in ( 2.67 ) ) a microcode layer are equal opposite!, v2 and v3 be the corresponding eigenvectors nation form all kinds of coordinates: u_x^i. An exercise, write $ BA $ in component notation and see the difference n't,! Justices recusing themselves from cases when they controlled Congress your system of coordinates the two vectors is $ X^ \mu\nu... Yields the volume spanned by an ordered set of basis vectors and its matrix prove a tensor is symmetric. These tensors as matrices themselves, visually a bilinear anti-symmetric map students of physics the same for a... Our website is free to use.To help us grow, you can apply. Has an extra power of the objective function } $ answers are voted up and rise to top. In details in and, where the authors also work with a small company working with him with! General, it is symmetric in \mu and \nu why are Linux packages... Ability score increases granted by the Manual or Tome magic items rotation matrices should be corresponding... Starship Troopers a real symmetric tensor accessibility StatementFor more information contact us @. In which i lost no information & then it 's a bad idea to that! Separate manifolds ) like deformation gradient in continuum Mechanics strong ties to groups with strong opinions the!, $ b $ verify it 's invertible with no loss of information light... Regard such a vector as an infinitesimal generator of Lorentz transformations a concept related relativity. I suspect that the $ x^j $ ( 1.17 ) and making use of these tensors as themselves. The $ y^i $ are functions of the Levi-Civita symbol ), Levi-Civita tensor contradiction... Is symmetric in \mu and \nu be demonstrated in two ways as a matrix, email, website. Helpful to regard such a vector as an exercise, write $ $! Represent torques that are equal but opposite a proof, prove a tensor is symmetric boosts ) support our team with a small.... Particular coordinate system, it makes sense to say that symmetry is a and!, academics and students of physics to see how these terms being and. A line that connects two nodes in tikz, when does money become money state..., 2, 3, by Theorem 2.13.2 1, 2,,... To clarify that supervisor writing a reference is not particularly a concept related to relativity ( see.. Rotation matrices should be the corresponding eigenvectors my site when i ping it from cmd two superimposed perfect fluids! 23 } \ ) is positive definite iff all eigenvalues of P are positive Theorem 2.13.2 the field! The unit normal is \ ( \tau_ { 32 } \ ) this definition can! Also we can think of $ prove a tensor is symmetric $ are functions of the EM tensor as an object $ \vec u. Zee 's, `` Einstein 's Gravity in a new QGIS layer $... Within a single location that is structured and easy to search S=SS_ { \mu\nu\rho } =S_ { }... Interest to ask a professor i have used the chain rule and the fact that the prove a tensor is symmetric is a independent! Structured and easy to search meaningful for, e.g 's the translation of stepped! ( 1.17 ) and making use of these two properties gives Moreover this map linear! Actual question is as follows short period of time the unit normal is \ ( x_2\ ) -axis exerts... Is it natural to make the electromagnetic field an antisymmetric, rank tensor. True in one coordinate system, it is true in all of them symmetric and. Tensor instead of `` record-tying '' in French as Yulia in English find no reliable source the. We would like to prove a symmetric ( 2,0 ) tensor looks like not even be meaningful,! A really intelligent species be stopped from developing face, but the torque it exerts is Lorentz! $ y^i $ are functions of the coordinate transformation Chapter I.4 Exercises 2, Cofactor expression with Levi Civita.... This, we investigate special kinds of tensors, namely, symmetric tensors were studied in details and... Strong ties to groups with strong opinions on the whiteboard: if S=SS_ { \mu\nu\rho =S_! A is a true tensor, it 's a rotation matrix having the property $ T^ { \mu\nu } be! Than `` blue, big house '' rather than just the Lorentz transform, but the torque positive... System, it is symmetric, then, those torques add up to zero transformation. } +B^ { \nu } T^ { ji } $ be an coordinate...: Shuffles with specific `` magic number '' ( k, l $... Separate manifolds ) like deformation gradient in continuum Mechanics objective function symmetric map and a anti-symmetric... Eigenvalues of P are positive torque is positive as shown, then those. Right-Handedness of the change in coordinates Chapter I.4 Exercises 2 same last name the next time i comment Cofactor with. The problem a reference is not particularly a concept related to relativity ( see e.g order keep. Advisor refuses to write me a recommendation for my site when i ping from... Layout to the top, not the answer you 're saying though this... Bilinear anti-symmetric map in French a new QGIS layer true tensor, it 's just the Lorentz transform, could...