Manifest covariance

<p>In <a href="/facts/General_relativity/jVsoIg8X">general relativity</a>, a manifestly covariant equation is one in which all expressions are <a href="/facts/Tensor/pNjFo5tV">tensors</a>.  The operations of addition, <a href="/facts/Tensor_multiplication/7K3pR1ap">tensor multiplication</a>, <a href="/facts/Tensor_contraction/fqz0zvWp">tensor contraction</a>, <a href="/facts/Raising_and_lowering_indices/cPsSVpgI">raising and lowering indices</a>, and <a href="/facts/Covariant_differentiation/sRGN0Xh6">covariant differentiation</a> may appear in the equation.  Forbidden terms include but are not restricted to <a href="/facts/Partial_derivatives/JWHsBkAu">partial derivatives</a>.  <a href="/facts/Tensor_density/XF4gqBCM">Tensor densities</a>, especially integrands and variables of integration, may be allowed in manifestly covariant equations if they are clearly weighted by the appropriate power of the <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of the metric.  
</p><p>Writing an equation in manifestly covariant form is useful because it guarantees <a href="/facts/General_covariance/KzTG2t9g">general covariance</a> upon quick inspection.  If an equation is manifestly covariant, and if it reduces to a correct, corresponding equation in <a href="/facts/Special_relativity/vsFFc63E">special relativity</a> when evaluated instantaneously in a <a href="/facts/Local_inertial_frame/WdBs8wgh">local inertial frame</a>, then it is usually the correct generalization of the special relativistic equation in general relativity.
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Manifest covariance open-in-new

Manifest covariance