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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Abstract index notation</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Tensor_index_notation" class="mw-redirect" title="Tensor index notation">tensor index notation</a>.</div>

<p><b>Abstract index notation</b> (also referred to as slot-naming index notation)<sup id="cite_ref-Thorne2017_1-0" class="reference"><a href="#cite_note-Thorne2017-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is a mathematical notation for <a href="Tensor" title="Tensor">tensors</a> and <a href="Spinor" title="Spinor">spinors</a> that uses indices to indicate their types, rather than their components in a particular basis.<sup id="cite_ref-Penrose2007_2-0" class="reference"><a href="#cite_note-Penrose2007-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The indices are mere placeholders, not related to any basis and, in particular, are non-numerical. Thus it should not be confused with the <a href="Ricci_calculus" title="Ricci calculus">Ricci calculus</a>. The notation was introduced by <a href="Roger_Penrose" title="Roger Penrose">Roger Penrose</a> as a way to use the formal aspects of the <a href="Einstein_summation_convention" class="mw-redirect" title="Einstein summation convention">Einstein summation convention</a> to compensate for the difficulty in describing <a href="Tensor_contraction" title="Tensor contraction">contractions</a> and <a href="Covariant_derivative" title="Covariant derivative">covariant differentiation</a> in modern abstract tensor notation, while preserving the explicit <a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">covariance</a> of the expressions involved.<sup id="cite_ref-Penrose1984_3-0" class="reference"><a href="#cite_note-Penrose1984-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> be a <a href="Vector_space" title="Vector space">vector space</a>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle V^{*}}</annotation>

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</math></span><img src="./5910e6a94f4f7ee2ee85ceed9dacef3eff7a6242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle V^{*}}" loading="lazy"></span> its <a href="Dual_space" title="Dual space">dual space</a>. Consider, for example, an order-2 <a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">covariant</a> tensor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in V^{*}\otimes V^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle h\in V^{*}\otimes V^{*}}</annotation>

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</math></span><img src="./07b12887bcd1b62326e123fce4f43818ffd05b51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.962ex; height:2.509ex;" alt="{\displaystyle h\in V^{*}\otimes V^{*}}" loading="lazy"></span>. Then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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</math></span><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> can be identified with a <a href="Bilinear_form" title="Bilinear form">bilinear form</a> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span>. In other words, it is a function of two arguments in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> which can be represented as a pair of <i>slots</i>:

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<annotation encoding="application/x-tex">{\displaystyle h=h(-,-).}</annotation>

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</math></span><img src="./3a45f9c6a14512f4cc3763415875973572373b32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.883ex; height:2.843ex;" alt="{\displaystyle h=h(-,-).}" loading="lazy"></span></dd></dl>

<p>Abstract index notation is merely a <i>labelling</i> of the slots with Latin letters, which have no significance apart from their designation as labels of the slots (i.e., they are non-numerical):

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<annotation encoding="application/x-tex">{\displaystyle h=h_{ab}.}</annotation>

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<p>A <a href="Tensor_contraction" title="Tensor contraction">tensor contraction</a> (or trace) between two tensors is represented by the repetition of an index label, where one label is contravariant (an <i>upper index</i> corresponding to the factor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span>) and one label is covariant (a <i>lower index</i> corresponding to the factor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}}">

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</math></span><img src="./5910e6a94f4f7ee2ee85ceed9dacef3eff7a6242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle V^{*}}" loading="lazy"></span>). Thus, for instance,

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<annotation encoding="application/x-tex">{\displaystyle t_{ab}{}^{b}}</annotation>

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<p>is the trace of a tensor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{ab}{}^{c}}">

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<annotation encoding="application/x-tex">{\displaystyle t=t_{ab}{}^{c}}</annotation>

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</math></span><img src="./91228a617a200192bdbbdc1ee7cfe5f214da17cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.529ex; height:2.676ex;" alt="{\displaystyle t=t_{ab}{}^{c}}" loading="lazy"></span> over its last two slots. This manner of representing tensor contractions by repeated indices is formally similar to the <a href="Einstein_summation_convention" class="mw-redirect" title="Einstein summation convention">Einstein summation convention</a>. However, as the indices are non-numerical, it does not imply summation: rather it corresponds to the abstract basis-independent trace operation (or <a href="Natural_pairing" class="mw-redirect" title="Natural pairing">natural pairing</a>) between tensor factors of type <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> and those of type <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle V^{*}}</annotation>

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</math></span><img src="./5910e6a94f4f7ee2ee85ceed9dacef3eff7a6242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle V^{*}}" loading="lazy"></span>.

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<div class="mw-heading mw-heading2"><h2 id="Abstract_indices_and_tensor_spaces">Abstract indices and tensor spaces</h2></div>

<p>A general homogeneous tensor is an element of a <a href="Tensor_product" title="Tensor product">tensor product</a> of copies of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle V^{*}}</annotation>

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</math></span><img src="./5910e6a94f4f7ee2ee85ceed9dacef3eff7a6242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle V^{*}}" loading="lazy"></span>, such as

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<annotation encoding="application/x-tex">{\displaystyle V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}.}</annotation>

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</math></span><img src="./fad0f977ecd7445c389e3d80fb2096bba3f258a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.496ex; height:2.509ex;" alt="{\displaystyle V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}.}" loading="lazy"></span></dd></dl>

<p>Label each factor in this tensor product with a Latin letter in a raised position for each contravariant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> factor, and in a lowered position for each covariant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle V^{*}}</annotation>

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</math></span><img src="./5910e6a94f4f7ee2ee85ceed9dacef3eff7a6242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle V^{*}}" loading="lazy"></span> position. In this way, write the product as

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<annotation encoding="application/x-tex">{\displaystyle V^{a}V_{b}V_{c}V^{d}V_{e}}</annotation>

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<p>or, simply

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<annotation encoding="application/x-tex">{\displaystyle V^{a}{}_{bc}{}^{d}{}_{e}.}</annotation>

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</math></span><img src="./005b56f79ca6ecbe7cb01ddcf40c40778854314f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.406ex; height:3.009ex;" alt="{\displaystyle V^{a}{}_{bc}{}^{d}{}_{e}.}" loading="lazy"></span></dd></dl>

<p>The last two expressions denote the same object as the first. Tensors of this type are denoted using similar notation, for example:

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<annotation encoding="application/x-tex">{\displaystyle h^{a}{}_{bc}{}^{d}{}_{e}\in V^{a}{}_{bc}{}^{d}{}_{e}=V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}.}</annotation>

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</math></span><img src="./e19ffec9f4f72d3f0346e5f3d0c963b75df6f91c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:43.375ex; height:3.009ex;" alt="{\displaystyle h^{a}{}_{bc}{}^{d}{}_{e}\in V^{a}{}_{bc}{}^{d}{}_{e}=V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Contraction">Contraction</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Tensor_contraction" title="Tensor contraction">Tensor contraction</a></div>

<p>In general, whenever one contravariant and one covariant factor occur in a tensor product of spaces, there is an associated <i>contraction</i> (or <i>trace</i>) map. For instance,

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Tr} _{12}:V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}\to V^{*}\otimes V\otimes V^{*}}</annotation>

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<p>is the trace on the first two spaces of the tensor product.<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Tr} _{15}:V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}\to V^{*}\otimes V^{*}\otimes V}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Tr} _{15}:V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}\to V^{*}\otimes V^{*}\otimes V}</annotation>

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</p><p>These trace operations are signified on tensors by the repetition of an index. Thus the first trace map is given by

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Tr} _{12}:h{}^{a}{}_{b}{}_{c}{}^{d}{}_{e}\mapsto h{}^{a}{}_{a}{}_{c}{}^{d}{}_{e}}</annotation>

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<p>and the second by

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Tr} _{15}:h{}^{a}{}_{b}{}_{c}{}^{d}{}_{e}\mapsto h{}^{a}{}_{b}{}_{c}{}^{d}{}_{a}.}</annotation>

</semantics>

</math></span><img src="./0344ba2be2e008f7d3c9291acb127f9c441ff1f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.594ex; height:3.009ex;" alt="{\displaystyle \mathrm {Tr} _{15}:h{}^{a}{}_{b}{}_{c}{}^{d}{}_{e}\mapsto h{}^{a}{}_{b}{}_{c}{}^{d}{}_{a}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Braiding">Braiding</h2></div>

<p>To any tensor product on a single vector space, there are associated <a href="Tensor_product#Tensor_powers_and_braiding" title="Tensor product">braiding maps</a>. For example, the braiding map

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<annotation encoding="application/x-tex">{\displaystyle \tau _{(12)}:V\otimes V\rightarrow V\otimes V}</annotation>

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</math></span><img src="./5e299c1a5846e2de48bc04f661bb1c8f203e1720.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.553ex; height:3.009ex;" alt="{\displaystyle \tau _{(12)}:V\otimes V\rightarrow V\otimes V}" loading="lazy"></span></dd></dl>

<p>interchanges the two tensor factors (so that its action on simple tensors is given by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{(12)}(v\otimes w)=w\otimes v}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \tau _{(12)}(v\otimes w)=w\otimes v}</annotation>

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</math></span><img src="./30a01db1ebdd0838858d6af1c30df0c78d08f166.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.344ex; height:3.176ex;" alt="{\displaystyle \tau _{(12)}(v\otimes w)=w\otimes v}" loading="lazy"></span>). In general, the braiding maps are in one-to-one correspondence with elements of the <a href="Symmetric_group" title="Symmetric group">symmetric group</a>, acting by permuting the tensor factors. Here, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{\sigma }}">

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<annotation encoding="application/x-tex">{\displaystyle \tau _{\sigma }}</annotation>

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</math></span><img src="./ea9d51dc2f1ee30df36e314a43ff5584b7df6e8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.189ex; height:2.009ex;" alt="{\displaystyle \tau _{\sigma }}" loading="lazy"></span> denotes the braiding map associated to the permutation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math></span><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"></span> (represented as a product of disjoint <a href="Cyclic_permutation" title="Cyclic permutation">cyclic permutations</a>).

</p><p>Braiding maps are important in <a href="Differential_geometry" title="Differential geometry">differential geometry</a>, for instance, in order to express the <a href="Bianchi_identity" class="mw-redirect" title="Bianchi identity">Bianchi identity</a>. Here let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> denote the <a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann tensor</a>, regarded as a tensor in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}\otimes V^{*}\otimes V^{*}\otimes V}">

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<annotation encoding="application/x-tex">{\displaystyle V^{*}\otimes V^{*}\otimes V^{*}\otimes V}</annotation>

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</math></span><img src="./2c23718e38ddf25a996b3efc68701fa117749cae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.222ex; height:2.509ex;" alt="{\displaystyle V^{*}\otimes V^{*}\otimes V^{*}\otimes V}" loading="lazy"></span>. The first Bianchi identity then asserts that

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<annotation encoding="application/x-tex">{\displaystyle R+\tau _{(123)}R+\tau _{(132)}R=0.}</annotation>

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</math></span><img src="./2726ae30234d12035dd13d9da9337d7e679ffba1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.868ex; height:3.009ex;" alt="{\displaystyle R+\tau _{(123)}R+\tau _{(132)}R=0.}" loading="lazy"></span></dd></dl>

<p>Abstract index notation handles braiding as follows. On a particular tensor product, an ordering of the abstract indices is fixed (usually this is a <a href="Lexicographic_order" title="Lexicographic order">lexicographic ordering</a>). The braid is then represented in notation by permuting the labels of the indices. Thus, for instance, with the Riemann tensor

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<annotation encoding="application/x-tex">{\displaystyle R=R_{abc}{}^{d}\in V_{abc}{}^{d}=V^{*}\otimes V^{*}\otimes V^{*}\otimes V,}</annotation>

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</math></span><img src="./43a3ca1f98e4e5577ca83a8cb26e8524ac40be60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:41.012ex; height:3.009ex;" alt="{\displaystyle R=R_{abc}{}^{d}\in V_{abc}{}^{d}=V^{*}\otimes V^{*}\otimes V^{*}\otimes V,}" loading="lazy"></span></dd></dl>

<p>the Bianchi identity becomes

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<annotation encoding="application/x-tex">{\displaystyle R_{abc}{}^{d}+R_{cab}{}^{d}+R_{bca}{}^{d}=0.}</annotation>

</semantics>

</math></span><img src="./a865d2f0eb304bc0a53829abd113290896e258c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.714ex; height:3.009ex;" alt="{\displaystyle R_{abc}{}^{d}+R_{cab}{}^{d}+R_{bca}{}^{d}=0.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Antisymmetrization_and_symmetrization">Antisymmetrization and symmetrization</h2></div>

<p>A general tensor may be antisymmetrized or symmetrized, and there is according notation.

</p><p>We demonstrate the notation by example. Let's antisymmetrize the type-(0,3) tensor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{abc}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \omega _{abc}}</annotation>

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</math></span><img src="./bfeef3f0b5e1004ee088b37d523838efae99e57a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.965ex; height:2.009ex;" alt="{\displaystyle \omega _{abc}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {S} _{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {S} _{3}}</annotation>

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</math></span><img src="./f7739a4f0628592d6e923ae31dac49b5ccd71d3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.347ex; height:2.509ex;" alt="{\displaystyle \mathrm {S} _{3}}" loading="lazy"></span> is the symmetric group on three elements.

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<annotation encoding="application/x-tex">{\displaystyle \omega _{[abc]}:={\frac {1}{3!}}\sum _{\sigma \in \mathrm {S} _{3}}{{\text{sgn}}(\sigma )}\omega _{\sigma (a)\sigma (b)\sigma (c)}}</annotation>

</semantics>

</math></span><img src="./ead5c0dc4d732e8c36e8c9d97ac0cb38b2b36346.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:32.961ex; height:6.843ex;" alt="{\displaystyle \omega _{[abc]}:={\frac {1}{3!}}\sum _{\sigma \in \mathrm {S} _{3}}{{\text{sgn}}(\sigma )}\omega _{\sigma (a)\sigma (b)\sigma (c)}}" loading="lazy"></span></dd></dl>

<p>Similarly, we may symmetrize:

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<annotation encoding="application/x-tex">{\displaystyle \omega _{(abc)}:={\frac {1}{3!}}\sum _{\sigma \in \mathrm {S} _{3}}\omega _{\sigma (a)\sigma (b)\sigma (c)}}</annotation>

</semantics>

</math></span><img src="./19fd61336c9d527b994f2b46a0f17c7067fcb49e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:26.815ex; height:6.843ex;" alt="{\displaystyle \omega _{(abc)}:={\frac {1}{3!}}\sum _{\sigma \in \mathrm {S} _{3}}\omega _{\sigma (a)\sigma (b)\sigma (c)}}" loading="lazy"></span></dd></dl>

<ul><li><a href="Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li>

<li><a href="Einstein_notation" title="Einstein notation">Einstein notation</a></li>

<li><a href="Index_notation" title="Index notation">Index notation</a></li>

<li><a href="Tensor" title="Tensor">Tensor</a></li>

<li><a href="Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li>

<li><a href="Raising_and_lowering_indices" class="mw-redirect" title="Raising and lowering indices">Raising and lowering indices</a></li>

<li><a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFKip_S._Thorne_and_Roger_D._Blandford2017" class="citation book cs1">Kip S. Thorne and Roger D. Blandford (2017). <i>Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics</i>. Princeton University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-69115902-7</bdi>.</cite></span>

</li>

<li id="cite_note-Penrose2007-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Penrose2007_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFRoger_Penrose2007" class="citation book cs1">Roger Penrose (2007). <i>The Road to Reality: A Complete Guide to the Laws of the Universe</i>. Vintage. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-67977631-4</bdi>.</cite></span>

</li>

<li id="cite_note-Penrose1984-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Penrose1984_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFRoger_Penrose_and_Wolfgang_Rindler1984" class="citation book cs1">Roger Penrose and Wolfgang Rindler (1984). <i>Spinors and Space-Time, Volume 1: Two-Spinor Calculus and Relativistic Fields</i>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-52133707-6</bdi>.</cite></span>

</li>

</ol></div></div>

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</style><div id="Tensors176" style="font-size:114%;margin:0 4em"><a href="Tensor" title="Tensor">Tensors</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i><a href="Glossary_of_tensor_theory" title="Glossary of tensor theory">Glossary of tensor theory</a></i></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Scope</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Mathematics" title="Mathematics">Mathematics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Coordinate_system" title="Coordinate system">Coordinate system</a></li>

<li><a href="Differential_geometry" title="Differential geometry">Differential geometry</a></li>

<li><a href="Dyadics" title="Dyadics">Dyadic algebra</a></li>

<li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li>

<li><a href="Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior calculus</a></li>

<li><a href="Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li>

<li><a href="Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div class="hlist"><ul><li><a href="Physics" title="Physics">Physics</a></li><li><a href="Engineering" title="Engineering">Engineering</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Computer_vision" title="Computer vision">Computer vision</a></li>

<li><a href="Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li>

<li><a href="Electromagnetism" title="Electromagnetism">Electromagnetism</a></li>

<li><a href="General_relativity" title="General relativity">General relativity</a></li>

<li><a href="Transport_phenomena" title="Transport phenomena">Transport phenomena</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul>

<li><a href="Einstein_notation" title="Einstein notation">Einstein notation</a></li>

<li><a href="Index_notation" title="Index notation">Index notation</a></li>

<li><a href="Multi-index_notation" title="Multi-index notation">Multi-index notation</a></li>

<li><a href="Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li>

<li><a href="Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li>

<li><a href="Tetrad_(index_notation)" class="mw-redirect" title="Tetrad (index notation)">Tetrad (index notation)</a></li>

<li><a href="Van_der_Waerden_notation" title="Van der Waerden notation">Van der Waerden notation</a></li>

<li><a href="Voigt_notation" title="Voigt notation">Voigt notation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tensor<br>definitions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></li>

<li><a href="Tensor_field" title="Tensor field">Tensor field</a></li>

<li><a href="Tensor_density" title="Tensor density">Tensor density</a></li>

<li><a href="Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li>

<li><a href="Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li>

<li><a href="Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li>

<li><a href="Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li>

<li><a href="Tensor_operator" title="Tensor operator">Tensor operator</a></li>

<li><a href="Tensor_bundle" title="Tensor bundle">Tensor bundle</a></li>

<li><a href="Two-point_tensor" title="Two-point tensor">Two-point tensor</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Operation_(mathematics)" title="Operation (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li>

<li><a href="Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li>

<li><a href="Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li>

<li><a href="Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li>

<li><a href="Lie_derivative" title="Lie derivative">Lie derivative</a></li>

<li><a href="Raising_and_lowering_indices" class="mw-redirect" title="Raising and lowering indices">Raising and lowering indices</a></li>

<li><a href="Symmetrization" title="Symmetrization">Symmetrization</a></li>

<li><a href="Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li>

<li><a href="Tensor_product" title="Tensor product">Tensor product</a></li>

<li><a href="Transpose" title="Transpose">Transpose</a> (2nd-order tensors)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br>abstractions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Affine_connection" title="Affine connection">Affine connection</a></li>

<li><a href="Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li>

<li><a href="Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li>

<li><a href="Connection_form" title="Connection form">Connection form</a></li>

<li><a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a></li>

<li><a href="Dimension" title="Dimension">Dimension</a></li>

<li><a href="Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li>

<li><a href="Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li>

<li><a href="Geodesic" title="Geodesic">Geodesic</a></li>

<li><a href="Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li>

<li><a href="Linear_map" title="Linear map">Linear map</a></li>

<li><a href="Manifold" title="Manifold">Manifold</a></li>

<li><a href="Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li>

<li><a href="Multivector" title="Multivector">Multivector</a></li>

<li><a href="Pseudotensor" title="Pseudotensor">Pseudotensor</a></li>

<li><a href="Spinor" title="Spinor">Spinor</a></li>

<li><a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li>

<li><a href="Vector_space" title="Vector space">Vector space</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notable tensors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li>

<li><a href="Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li>

<li><a href="Metric_tensor" title="Metric tensor">Metric tensor</a></li>

<li><a href="Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li>

<li><a href="Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li>

<li><a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li>

<li><a href="Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li>

<li><a href="Weyl_tensor" title="Weyl tensor">Weyl tensor</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Physics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li>

<li><a href="Angular_momentum#Angular_momentum_in_relativistic_mechanics" title="Angular momentum">Angular momentum tensor</a></li>

<li><a href="Spin_tensor" title="Spin tensor">Spin tensor</a></li>

<li><a href="Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li>

<li><a href="Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li>

<li><a href="Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li>

<li><a href="Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li>

<li><a href="Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li>

<li><a href="Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">Metric tensor (GR)</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematician" title="Mathematician">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li>

<li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li>

<li><a href="Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li>

<li><a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li>

<li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li>

<li><a href="Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li>

<li><a href="Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a></li>

<li><a href="Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a></li>

<li><a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li>

<li><a href="Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Jan Arnoldus Schouten</a></li>

<li><a href="Woldemar_Voigt" title="Woldemar Voigt">Woldemar Voigt</a></li>

<li><a href="Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Roger_Penrose53" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Roger_Penrose53" style="font-size:114%;margin:0 4em"><a href="Roger_Penrose" title="Roger Penrose">Roger Penrose</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Books</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><i><a href="The_Emperor's_New_Mind" title="The Emperor's New Mind">The Emperor's New Mind</a></i> (1989)</li>

<li><i><a href="Shadows_of_the_Mind" title="Shadows of the Mind">Shadows of the Mind</a></i> (1994)</li>

<li><i><a href="The_Road_to_Reality" title="The Road to Reality">The Road to Reality</a></i> (2004)</li>

<li><i><a href="Cycles_of_Time" title="Cycles of Time">Cycles of Time</a></i> (2010)</li>

<li><i><a href="Fashion%2C_Faith%2C_and_Fantasy_in_the_New_Physics_of_the_Universe" title="Fashion, Faith, and Fantasy in the New Physics of the Universe">Fashion, Faith, and Fantasy in the New Physics of the Universe</a></i> (2016)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coauthored books</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><span class="wrap"> <i><a href="The_Nature_of_Space_and_Time" title="The Nature of Space and Time">The Nature of Space and Time</a></i> (with <a href="Stephen_Hawking" title="Stephen Hawking">Stephen Hawking</a>) (1996) </span></li>

<li><span class="wrap"> <i><a href="The_Large%2C_the_Small_and_the_Human_Mind" title="The Large, the Small and the Human Mind">The Large, the Small and the Human Mind</a></i> (with <a href="Abner_Shimony" title="Abner Shimony">Abner Shimony</a>, <a href="Nancy_Cartwright_(philosopher)" title="Nancy Cartwright (philosopher)">Nancy Cartwright</a> and Stephen Hawking) (1997) </span></li>

<li><span class="wrap"> <i>White Mars or, The Mind Set Free</i> (with <a href="Brian_W._Aldiss" class="mw-redirect" title="Brian W. Aldiss">Brian W. Aldiss</a>) (1999) </span></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Twistor_theory" title="Twistor theory">Twistor theory</a></li>

<li><a href="Spin_network" title="Spin network">Spin network</a></li>

<li><a href="Black_hole_bomb" title="Black hole bomb">Black hole bomb</a></li>

<li><a href="Spacetime" title="Spacetime">Geometry of spacetime</a></li>

<li><a href="Cosmic_censorship_hypothesis" title="Cosmic censorship hypothesis">Cosmic censorship</a></li>

<li><a href="Weyl_curvature_hypothesis" title="Weyl curvature hypothesis">Weyl curvature hypothesis</a></li>

<li><a href="Contributors_to_general_relativity" class="mw-redirect" title="Contributors to general relativity">Penrose inequalities</a></li>

<li><a href="Penrose_interpretation" title="Penrose interpretation">Penrose interpretation of quantum mechanics</a></li>

<li><a href="Moore%E2%80%93Penrose_inverse" title="Moore–Penrose inverse">Moore–Penrose inverse</a></li>

<li><a href="Newman%E2%80%93Penrose_formalism" title="Newman–Penrose formalism">Newman–Penrose formalism</a></li>

<li><a href="Penrose_diagram" title="Penrose diagram">Penrose diagram</a></li>

<li><a href="Penrose%E2%80%93Hawking_singularity_theorems" title="Penrose–Hawking singularity theorems">Penrose–Hawking singularity theorems</a></li>

<li><a href="Riemannian_Penrose_inequality" title="Riemannian Penrose inequality">Riemannian Penrose inequality</a></li>

<li><a href="Penrose_process" title="Penrose process">Penrose process</a></li>

<li><a href="Penrose_tiling" title="Penrose tiling">Penrose tiling</a></li>

<li><a href="Penrose_triangle" title="Penrose triangle">Penrose triangle</a></li>

<li><a href="Penrose_stairs" title="Penrose stairs">Penrose stairs</a></li>

<li><a href="Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li>

<li><a href="Penrose_transform" title="Penrose transform">Penrose transform</a></li>

<li><a href="Terrell_rotation" title="Terrell rotation">Penrose–Terrell effect</a></li>

<li><a href="Orchestrated_objective_reduction" title="Orchestrated objective reduction">Orchestrated objective reduction</a>/<a href="Penrose%E2%80%93Lucas_argument" title="Penrose–Lucas argument">Penrose–Lucas argument</a></li>

<li><a href="Free-orbit_experiment_with_laser_interferometry_X-rays" title="Free-orbit experiment with laser interferometry X-rays">FELIX experiment</a></li>

<li><a href="Trapped_surface" title="Trapped surface">Trapped surface</a></li>

<li><a href="Andromeda_paradox" class="mw-redirect" title="Andromeda paradox">Andromeda paradox</a></li>

<li><a href="Conformal_cyclic_cosmology" title="Conformal cyclic cosmology">Conformal cyclic cosmology</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Lionel_Penrose" title="Lionel Penrose">Lionel Penrose</a> (father)</li>

<li><a href="Oliver_Penrose" title="Oliver Penrose">Oliver Penrose</a> (brother)</li>

<li><a href="Jonathan_Penrose" title="Jonathan Penrose">Jonathan Penrose</a> (brother)</li>

<li><a href="Shirley_Hodgson" title="Shirley Hodgson">Shirley Hodgson</a> (sister)</li>

<li><a href="John_Beresford_Leathes" title="John Beresford Leathes">John Beresford Leathes</a> (grandfather)</li>

<li><a href="Illumination_problem" title="Illumination problem">Illumination problem</a></li>

<li><a href="Quantum_mind" title="Quantum mind">Quantum mind</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-01-30" href="https://en.wikipedia.org/wiki/?title=Abstract_index_notation&oldid=1272901227">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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