What We Do Know

When we have a wave in 3D space,

$\cfrac { \partial ^{ 2 }\psi  }{ \partial \, { t }^{ 2 } } ={ c }^{ 2 }\cfrac { \partial ^{ 2 }\psi  }{ \partial \, { x }^{ 2 } }$

A valid solution is,

$\psi =\psi _{ o }e^{ i\left( wt+kx \right)  }=\psi _{ o }e^{ iwt }e^{ ikx }$

where a particle at $x=x_o$ experiences $e^{ iwt }$.  As each particle has an exclusive solution to $x$ then each particle experiences $t$ differently.

When we have a wave in 3D time,

$\cfrac { \partial ^{ 2 }\psi  }{ \partial \, { x }^{ 2 } } ={ c }^{ 2 }_{ t }\cfrac { \partial ^{ 2 }\psi  }{ \partial \, { t }^{ 2 } }$

A similar solution is,

$\psi =\psi _{ o }e^{ i\left( wx+kt \right)  }=\psi _{ o }e^{ iwx }e^{ ikt }$

where a particle in time at $t=t_o$ experiences $e^{ iwx }$.  As each particle has an exclusive solution to $t$ then each particle experiences $x$ differently.  In this case, time standstill and space $x$ passes by.

If

$\cfrac{\partial\,x}{\partial\,t}=\cfrac{1}{\left(\cfrac{\partial\,t}{\partial\,x}\right)}$

then,

$c=\cfrac{1}{c_t}$

assuming all space and time dimensions are equivalent, then both wave equations are equivalent.

What if not all space and time dimensions are equivalent?

What if?