If we look again at the conservation of energy equation,
\( { v }_{ t }^{ 2 }+{ v }_{ s }^{ 2 }={ c }^{ 2 }\)
what happens when \( { v }_{ s }>{ c }\),
\( { v }_{ t }^{ 2 }= -{ V }^{ 2 }\) where \({ V }^{ 2 }={ v }_{ s }^{ 2 }-{ c }^{ 2 }\)
\( { v }_{ t }^{ ' }=iV\)
\(\nwarrow \quad +\quad \nearrow \quad +\quad \swarrow \quad=\quad \upuparrows \)
\( { v }_{ t }\) \( { v }_{ s }\) \( i{ v }_{ t }^{ ' }\) \( {c }\)
A complex component develops in \({ v }_{ t } \) that cancels that part of \({ v }_{ s }\) that is greater that \({ c }\). The use of \( i{ v }_{ t }^{ ' }\) is to illustrate that \( { v }_{ t }\) and \( i{ v }_{ t }^{ ' }\) are not of the same magnitude.
Similarly, if we develop a complex component in \({ v }_{ s }\) we will then be able to cancel part of \({ v }_{ t } \) and given a great enough complex component in \({ v }_{ s }\), we will be able to make \({ v }_{ t } \) negative. \( { v }_{ t }\), \( { v }_{ s }\) are dummy variables we can swap them and we have
\(\nwarrow \quad +\quad \nearrow \quad +\quad \swarrow \quad=\quad \upuparrows \)
\( { v }_{ s }\) \( { v }_{ t }\) \( i{ v }_{ s }^{ ' }\) \( {c }\)
Remember that \( {c }\) is a constant. If we are able to generate \( i{ v }_{ s }^{ ' }\) we can then slow \( { v }_{ t }\) and even reverse \( { v }_{ t }\).
