remove big space between equations in split environment

Question

I am new to latex and struggling to remove big space between the equations and between equation and text. I am using \documentclass{extarticle} and the equations I am writing are big which looks like

\begin{equation}

\begin{split}

Long equations taking more than half page

\end{split}

\end{equation}

I think that latex is taking all the equations as one piece and that's why when half page is left, the equations are shifted to next page. (I am using split command, because it aligns the equations easily using '&') Please tell me how to get rid of this problem.

The actual code is

 \section{Diffeomorphism}
$L_o = m(\dot{x}^2)^\frac{1}{2}$\\[10pt]
$\dot{x}^2 = \dot{x_\mu}\dot{x^\mu} = \dot{x_0}\dot{x^0}+\dot{x_1}\dot{x^1}+.....................+\dot{x_{D-1}}\dot{x^{D-1}}$\\[15pt]
Action $S = \int_{-\infty}^{\infty} d\tau \; m(\dot{x}^2)^\frac{1}{2}$\\[10pt]
Diffeomorphism\\
$\tau \rightarrow \tau' = f(\tau) \simeq \tau -\epsilon(\tau)$\\
$f(\tau) = finite\; at\; \tau =0$\\
$f(\tau) \rightarrow 0$ as $\tau \rightarrow \pm \infty$\\
Action remains invariant under Diffeomorphism Transformations. We can prove it as follows:\\
\begin{equation}
\begin{split}
S' & = \int_{-\infty}^{\infty} d\tau' \; m\left[\left(\frac{dx^\mu}{d\tau'}\right)\left(\frac{dx_\mu}{d\tau'}\right)\right]^\frac{1}{2}\\[10pt]
&=\int_{-\infty}^{\infty} \frac{d\tau'}{d\tau}d\tau \; m\left[\left(\frac{dx^\mu}{d\tau}\right)\left(\frac{d\tau}{d\tau'}\right)\left(\frac{dx_\mu}{d\tau}\right)\left(\frac{d\tau}{d\tau'}\right)\right]^\frac{1}{2}\\[10pt]
&=\int_{-\infty}^{\infty} d\tau \left(\frac{d\tau'}{d\tau}\right)\left(\frac{d\tau}{d\tau'}\right) \; m\left[\left(\frac{dx^\mu}{d\tau}\right)\left(\frac{dx_\mu}{d\tau}\right)\right]^\frac{1}{2}\\[10pt]
&= \int_{-\infty}^{\infty} d\tau \; m(\dot{x}^2)^\frac{1}{2}\\[10pt]
&=S
\end{split}
\end{equation} 

The output is

Answer 1

Short answer, no, you can't break an equation inside an split environment. One you can use is align. Try using the allowdisplaybreaks command.

\begingroup
\allowdisplaybreaks
\begin{align}
    ....
\end{align}
\endgroup

I hope it helps

Answer 2

I don't know if I have given to you a good answer and I have followed your request but I have written everything on one page. Here I add my MWE example with some additional packages like parskip to justify the text and I have improved some commands.

\documentclass{extarticle}
\usepackage{mathtools,amssymb}
\usepackage{parskip}
\begin{document}
\section{Diffeomorphism}
$L_0 = m(\dot{x}^2)^\frac{1}{2}$.
\vskip2pt
\[\dot{x}^2 = \dot{x_\mu}\dot{x}^\mu = \dot{x_0}\dot{x}^0+\dot{x_1}\dot{x}^1+\cdots\cdots\cdots+\dot{x}_{D-1}\dot{x}^{D-1}.\]
\vskip2pt
\textbf{Action} $S = \displaystyle \int_{-\infty}^{\infty} d\tau \; m(\dot{x}^2)^\frac{1}{2}$.
\vskip2pt
\textbf{Diffeomorphism}
$\tau \rightarrow \tau' = f(\tau) \simeq \tau -\epsilon(\tau)$.

$f(\tau) =\text{ finite at }\tau =0$.

$f(\tau) \rightarrow 0$ as $\tau \rightarrow \pm \infty$.

Action remains invariant under \textsl{Diffeomorphism Transformations}. We can prove it as follows:
\vspace{-.2cm}
\begin{equation}
\begin{split}
S' & = \int_{-\infty}^{\infty} d\tau' \; m\left[\left(\frac{dx^\mu}{d\tau'}\right)\left(\frac{dx_\mu}{d\tau'}\right)\right]^\frac{1}{2}\\
&=\int_{-\infty}^{\infty} \left(\frac{d\tau'}{d\tau}\right) d\tau \; m\left[\left(\frac{dx^\mu}{d\tau}\right)\left(\frac{d\tau}{d\tau'}\right)\left(\frac{dx_\mu}{d\tau}\right)\left(\frac{d\tau}{d\tau'}\right)\right]^\frac{1}{2}\\
&=\int_{-\infty}^{\infty} d\tau \left(\frac{d\tau'}{d\tau}\right)\left(\frac{d\tau}{d\tau'}\right) \; m\left[\left(\frac{dx^\mu}{d\tau}\right)\left(\frac{dx_\mu}{d\tau}\right)\right]^\frac{1}{2}\\
&= \int_{-\infty}^{\infty} d\tau \; m(\dot{x}^2)^\frac{1}{2}\\
&=S
\end{split}
\end{equation} 
\end{document}