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In the Alcubierre metric, why is there a beta with subscript multiplied by a beta with superscript? I know beta with subscript is the shift vector, but what is the difference between the two? $$\text ds^2 = -(\alpha^2 - \beta_i \beta^i) \text dt^2 + 2 \beta_i \text dx^i \text dt + \gamma_{ij} \text dx^i \text dx^j$$

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  • $\begingroup$ Please note that Mathjax is the site standard for mathematical expressions and images of math or text is strongly discouraged. $\endgroup$ – StephenG Jun 13 at 0:18
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This is how the dot product is defined for covariant and contravariant vectors (without explicitly inserting the metric) i.e, with the metric, the dot product would look like $$\beta \cdot \beta = \beta_i \beta^i = \gamma_{ij} \beta^j \beta^i$$

Note that $$\beta_i = \gamma_{ij} \beta^j $$ and $$\beta^i = \gamma^{ij}\beta_{j}$$

The metric $\gamma_{ij}$ raises and lowers indices in the equation you have mentioned.

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  • $\begingroup$ So there are two different shift vectors? $\endgroup$ – user345249 Jun 13 at 0:28
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    $\begingroup$ No. One is in covariant form and the other is in contravariant form. They are still the same vector. This is how dot products are done in general relativity/differential geometry. $\endgroup$ – joseph h Jun 13 at 0:51
  • $\begingroup$ I see. Would the distinction between the covariant and contra-variant components even matter in this case, since the spatial coordinates are orthogonal? $\endgroup$ – user345249 Jun 19 at 1:17
  • $\begingroup$ @user345249 even in orthogonal frames, the switch between covariant to contravariant switches the sign of the time-like components (or all the signs of the space-like components, depending on the signature convention), so if you don't keep track of them consistently, you might miss the correct sign for a coordinate-measured quantity $\endgroup$ – lurscher Jun 29 at 0:45
  • $\begingroup$ Does the time-like component sign-switch still matter in this case, even though they are 3-vectors? $\endgroup$ – user345249 Jul 14 at 16:13
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The sub and superscripts represent contra and covariant vectors. We can (sometimes) think of these vectors as column vectors (contravariant) and row vectors (covariant or covectors). An intuitive way to understand the difference between the two is to consider how each transforms under a change of basis. A covector transforms in the same way as its basis transforms, whereas a contravariant vector transforms inversely to how its basis transforms. An example would be the change of units from m to cm on a contravariant position vector, doing that transformation multiples all components of the position vector by 100 whereas the basis is changing by a factor of 1/100. A covariant position vector (sometimes called a dual vector) would have its components divided by 100 in a change of units from m to cm just like its basis. To your question, the contra and covariant indices on the shift vector imply a sum over spatial coordinates only since they are latin scripts and not Greek spacetime indices, doing so effectively gives the magnitude of the given component for a given spatial coordinate. As to its more physical meaning, I can't help there sadly.

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I believe the indices are raised and lowered using the spatial $\gamma_{ij}$ as metric.

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