Elektrisches Dipolmoment
![{\displaystyle p=(E,\mathbf {p} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73d9f333a8b7f96d22237fca1ec97f42a4e7412c)
Elektrische Stromdichte
![{\displaystyle j=(p,\mathbf {j} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb47953318457cfd6444d38a7393818c3390224)
Magnetisches Vektorpotential
![{\displaystyle A=(V,\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59d72c94f22c617e90355a2127db5b927c728179)
Ableitung
![{\displaystyle \partial _{\mu }\equiv \left({\frac {\partial }{\partial \,x^{\mu }}}\right)_{\mu }=\left({\frac {\partial }{\partial \,t}},\mathbf {\nabla } \right)_{\mu }=\left({\frac {\partial }{\partial \,t}},{\frac {\partial }{\partial \,x}},{\frac {\partial }{\partial \,y}},{\frac {\partial }{\partial \,z}}\right)_{\mu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/346de8ad1ec1a57c8a030bc2d545659c1d13d240)
![{\displaystyle g_{\mu \nu }={\begin{cases}1&{\text{if }}\mu =\nu =0\\-\delta _{\mu }^{\nu }&{\text{if }}(\mu \neq 0)\lor (\nu \neq 0)\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5959ba1d32c444a53ff638745ee3d492e20d1f5)
![{\displaystyle g_{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b40769e2bfa60e3993810a2e704cdf31db05c7)
Hoch- und Tiefstellen von Indizes:
![{\displaystyle a_{\mu }=g_{\mu \nu }\,a^{\nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6b005398507c40efdc9ed8fa46c297d67f14b49)
![{\displaystyle a_{0}=a^{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8eca0477132858b496ebe61ba856d26ccc4900d)
![{\displaystyle a_{i}=-a^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41e257a195416b6195a7b467cc1ad94aec82a39e)
![{\displaystyle \delta _{\mu }^{\nu }={\begin{cases}0{\text{ if }}\mu \neq \nu \\1{\text{ if }}\mu =\nu \end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0465b66024977bcf4799578260258208697d0935)
![{\displaystyle \delta _{\mu \nu }=g_{\mu \xi }\,\delta _{\nu }^{\xi }=g_{\nu \mathrm {o} }\,g_{\mu \xi }\,\delta ^{\xi \mathrm {o} }=\delta ^{\mu \nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77bb9534b38519966cc68908991b113f17aaea1b)
![{\displaystyle a\cdot b=a^{0}\,b^{0}-\mathbf {a} \cdot \mathbf {b} =g_{\mu \nu }\,a^{\mu }\,b^{\nu }=a_{\mu }\,b^{\nu }=a^{\mu }\,b_{\nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dbfffb239e4e2661d870551d80139ffb25dc991)
- Beispiele
![{\displaystyle p\cdot p=p_{\mu }\,p^{\mu }=(E,\mathbf {p} )\cdot (E,\mathbf {p} )=E^{2}-\mathbf {p} ^{2}=m^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b974a6da44447d085e7b3595436d357670845d3)
(wg. Summenkonvention)
d’Alembert-Operator:
![{\displaystyle \square =\partial ^{2}=\partial _{\mu }^{\mu }={\dfrac {\partial ^{2}}{\partial (ct)^{2}}}-{\dfrac {\partial ^{2}}{\partial x^{2}}}-{\dfrac {\partial ^{2}}{\partial y^{2}}}-{\dfrac {\partial ^{2}}{\partial z^{2}}}={\dfrac {\partial ^{2}}{c^{2}\partial t^{2}}}-{\boldsymbol {\nabla }}^{2}={\dfrac {\partial ^{2}}{c^{2}\partial t^{2}}}-{\boldsymbol {\Delta }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c87f00645afc015ec5d8ac5fe08c4b0dae90d4e1)
- Kontravariante Transformation
Transformation von Vektor von Inertialsystem (Basis) nach
![{\displaystyle r'^{\mu }=\left({\frac {\partial \,{\mathcal {S'}}^{\mu }}{\partial \,{\mathcal {S}}^{\nu }}}\right)\,r^{\nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bc2051211ba0379d1635277694447341a80a95f)
- Kovariante Transformation
Transformation von Gradientvektor von Inertialsystem (Basis) nach
![{\displaystyle {\frac {\partial \,\phi }{\partial \,{\mathcal {S'}}^{\mu }}}=\left({\frac {\partial \,{\mathcal {S}}^{\mu }}{\partial \,{\mathcal {S'}}^{\nu }}}\right)\,{\frac {\partial \,\phi }{\partial \,{\mathcal {S}}^{\nu }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7da6ed2090950f9b40064a2b2900bcce2f31997c)
![{\displaystyle {\begin{pmatrix}ct'\\r'^{i}\end{pmatrix}}={\begin{pmatrix}\gamma &-\gamma \beta _{i}\\-\gamma \beta _{j}&\delta _{ij}+(\gamma -1){\dfrac {\beta _{i}\beta _{j}}{\beta ^{2}}}\\\end{pmatrix}}{\begin{pmatrix}ct\\r^{i}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91a46ca8479532222e8289564440ace7e6913497)
mit
![{\displaystyle \gamma ={\frac {1}{\left(1-\left(\mathbf {\beta } \right)^{2}\right)^{\frac {1}{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce60ed4013bd0262a8ae9144035779c28f467715)
![{\displaystyle \mathbf {\beta } ^{i}={\frac {\mathbf {v} ^{i}}{c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f98320c9f8546c6590ef33f5e53b442ff8c6656)
![{\displaystyle c=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3467f9e219a5ea38a30da5c3a02c2c23f61a79)
- als Tensor
![{\displaystyle r'^{\mu }={\Lambda ^{\mu }}_{\nu }\,r^{\nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67ef794eb812c155e83c8a800a5f67a0231e2aa5)
mit
![{\displaystyle {\Lambda ^{\mu }}_{\nu }={\frac {\partial \,{\mathcal {S}}^{\mu }}{\partial \,{\mathcal {S'}}^{\nu }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8d60145da7a3fc6f5daf5637ecaea225270bb67)
- als Matrix
![{\displaystyle \left[{\Lambda ^{\mu }}_{\nu }\right]_{\mathcal {S}}={\begin{pmatrix}\gamma &-\beta _{1}\,\gamma &-\beta _{2}\,\gamma &-\beta _{3}\,\gamma \\-\beta _{1}\,\gamma &\delta _{11}+\left(\gamma -1\right)\,{\dfrac {\beta _{1}\,\beta _{1}}{(\beta )^{2}}}&\delta _{12}+\left(\gamma -1\right)\,{\dfrac {\beta _{1}\,\beta _{2}}{(\beta )^{1}}}&\delta _{13}+\left(\gamma -1\right)\,{\dfrac {\beta _{1}\,\beta _{3}}{(\beta )^{2}}}\\-\beta _{2}\,\gamma &\delta _{21}+\left(\gamma -1\right)\,{\dfrac {\beta _{2}\,\beta _{1}}{(\beta )^{2}}}&\delta _{22}+\left(\gamma -1\right)\,{\dfrac {\beta _{2}\,\beta _{2}}{(\beta )^{2}}}&\delta _{23}+\left(\gamma -1\right)\,{\dfrac {\beta _{2}\,\beta _{3}}{(\beta )^{2}}}\\-\beta _{3}\,\gamma &\delta _{31}+\left(\gamma -1\right)\,{\dfrac {\beta _{3}\,\beta _{1}}{(\beta )^{2}}}&\delta _{32}+\left(\gamma -1\right)\,{\dfrac {\beta _{3}\,\beta _{2}}{(\beta )^{2}}}&\delta _{33}+\left(\gamma -1\right)\,{\dfrac {\beta _{3}\,\beta _{3}}{(\beta )^{2}}}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe6da3fbbb320dde56d011dd2dae0fc4f696f3c)
mit
![{\displaystyle \Lambda _{00}=\gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb9a3dec0e915bf97513b95f08997d5c5dbbff16)
![{\displaystyle \Lambda _{0i}=\Lambda _{i0}=-\gamma \beta _{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a31edb5ef0fe78acde6342e97896b781056c450)
![{\displaystyle \Lambda _{ij}=\Lambda _{ji}=\delta _{ij}+(\gamma -1){\dfrac {\beta _{i}\beta _{j}}{\beta ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74abfbb1c2cbd7927a01a0a9fc02a98f97434a0d)
![{\displaystyle {T'}_{l'\ldots n'}^{i'\ldots k'}={\dfrac {\partial {x'}^{i'}}{\partial {x}^{i}}}\cdots {\dfrac {\partial {x'}^{k'}}{\partial {x}^{k}}}\ {\dfrac {\partial {x}^{l}}{\partial {x'}^{l'}}}\cdots {\dfrac {\partial {x}^{n}}{\partial {x'}^{n'}}}\ T_{l\ldots n}^{i\ldots k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2da4743726eee2fcbd7c67f37ca76c70f4a257eb)
- Beispiel
![{\displaystyle {\delta '}_{i}^{j}={\dfrac {\partial {x'}^{i}}{\partial {x}^{k}}}\ {\dfrac {\partial {x}^{l}}{\partial {x'}^{j}}}\ {\delta }_{k}^{l}={\dfrac {\partial {x'}^{i}}{\partial {x}^{k}}}\ {\dfrac {\partial {x}^{k}}{\partial {x'}^{j}}}={\delta }_{i}^{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31dabe6fd6c9b997d8125be4f3745277fc6b0c68)
![{\displaystyle \varepsilon ^{i_{1}\dots i_{n}}=\prod _{1\leq p<q\leq n}{\frac {i_{p}-i_{q}}{p-q}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b47ae101e742161a6da0f33241c6442c8e043e6c)
Anwendung:
![{\displaystyle (\mathbf {b} \times \mathbf {c} )^{i}=\epsilon ^{ijk}\,b^{j}\,c^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d42bc4aabf7e106387cb1dc84719c80cfec7d41)
![{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )^{i}=\epsilon ^{ijk}\,a^{i}\,b^{j}\,c^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e6483d0cd4a8ec316d203ecb560cf543d28dcaf)
![{\displaystyle \det \mathbf {A} =\epsilon _{i_{1}\,i_{2}\,\ldots \,i_{n}}\,A_{1\,i_{1}}\,A_{2\,i_{2}}\,\ldots \,A_{n\,i_{n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29f0665d4a3dbe49a5d028001126ea22a7ce9bf0)
- Beispiel
![{\displaystyle i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20) |
![{\displaystyle j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0) |
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![{\displaystyle i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20) |
![{\displaystyle j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0) |
![{\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40) |
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![{\displaystyle i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20) |
![{\displaystyle j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0) |
![{\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40) |
![{\displaystyle l}](https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac) |
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