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Let ##j^{\mu}(x)## be a Lorentz 4-vector field in Minkowski spacetime and let ##\Sigma## be a 3-dimensional spacelike hypersurface with constant time of some Lorentz frame. From those I can construct the quantity
$$Q=\int_{\Sigma} dS_{\mu}j^{\mu}$$
where
$$dS_{\mu}=d^3x n_{\mu}$$
and ##n_{\mu}## is the unit timelike vector normal to ##\Sigma##. The quantity ##Q## is a Lorentz scalar. Since ##j^{\mu}## is a Lorentz vector, it follows that ##dS_{\mu}## must also be a Lorentz vector. But ##n_{\mu}## is also a Lorentz vector, so ##dS_{\mu}## can be a Lorentz vector only if ##d^3x## is a Lorentz scalar. Yet, ##d^3x## is not a Lorentz scalar, leading to a contradiction.
Where is the error?
There is a similar "paradox" with 4-momentum defined as
$$P^{\nu}=\int_{\Sigma} dS_{\mu}T^{\mu\nu}$$
$$Q=\int_{\Sigma} dS_{\mu}j^{\mu}$$
where
$$dS_{\mu}=d^3x n_{\mu}$$
and ##n_{\mu}## is the unit timelike vector normal to ##\Sigma##. The quantity ##Q## is a Lorentz scalar. Since ##j^{\mu}## is a Lorentz vector, it follows that ##dS_{\mu}## must also be a Lorentz vector. But ##n_{\mu}## is also a Lorentz vector, so ##dS_{\mu}## can be a Lorentz vector only if ##d^3x## is a Lorentz scalar. Yet, ##d^3x## is not a Lorentz scalar, leading to a contradiction.
Where is the error?
There is a similar "paradox" with 4-momentum defined as
$$P^{\nu}=\int_{\Sigma} dS_{\mu}T^{\mu\nu}$$
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