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Suppose, the length, breadth, & height of a box are all equal, it has the shape of a cube and volume $V$. Now, if the box travels at uniform speed $v \gt 0$, then the length $L_0$ will become $L$,

$$ L = L_0 \, \sqrt{ 1 - \frac{ v^2 }{ c^2 } }. $$

Has the volume of the box changed to $\sqrt{ 1 - \frac{ v^2 }{ c^2 } } \, V$, or to $\left( \sqrt{ 1 - \frac{ v^2 }{ c^2 } } \right)^{\! \! 3}\, V$ ?

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Length contraction occurs along the direction/dimension of motion (relatively). So your dimension of box should be $0.51,1,1 (\text{metres})$. The other dimensions remain unaltered.

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    $\begingroup$ It's worth adding to this, into hammer home the point that "four dimensional things are invariant", that since time dialation has that factor of $\frac{1}{\gamma}$ in it, the 4-volume of that box, over some time interval $\Delta T$ is invariant under a lorentz transformation, $\endgroup$ Nov 9 '21 at 17:36
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"Suppose, the length, breadth, & height of a box are all equal, it has the shape of a cube and volume $V$."

Alright.

"Now, if the box travels at uniform speed $v \gt 0$,"

... To be specific, let's say that the box travels at speed $v$ with respect to a suitably large touchpad (or floor sensor array); or in other words: the touchpad travels at speed $v$ with respect to the box; or in other words: box and touchpad are sliding along each other, straight and uniformly, at speed $v$. ...

"then the length $L_0$ will become $L = L_0 \, \sqrt{1 - (v/c)^2}$"

It's a strange, arguably even inappropriate choice of words to say that "length $L_0$ will become [length] $L \lt L_0$".

For the described setup, the "length contraction" formula $L = L_0 \, \sqrt{1 - (v/c)^2}$ applies instead to $L_0$ symbolizing the (constant!) length of the box, and $L$ symbolizing the (constant!) distance between certain pairs of sensor elements of the touch pad; namely of any pair, e.g. the touchpad elements $J$ and $K$, which are paired up by satisfying the condition that $J$'s instant of being passed by the leading edge of the box was simultaneous to $K$'s instant of being passed by the trailing edge of the box.

Length $L$ can consequently be called the length of any so-called simultaneity projection of the box into the (inertial system of) the touch pad.

Length contraction is a relation between the distance between one pair (of "ends"), and the distance between certain other pairs (of "ends") which represent a simultaneity projection of the former pair.

"Has the volume of the box changed [...] ?"

Not at all: The box remains as specified: with its length, breadth, & height all equal, $L_0$.

The length value $L_0 \, \sqrt{1 - (v/c)^2}$ instead characterizes certain pairs of elements of the touchpad, which are identified by the simultaneity of one end indicating the passage of the leading edge of the box, and the other end indicating the passage of the trailing edge of the box.

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