![ $p+\rho$ as a function of the redshift $z$ and the wormhole radius $r$.[]{data-label="f7"}](7.pdf "fig:") ![ $p+\rho$ as a function of the redshift $z$ and the wormhole radius $r$.[]{data-label="f7"}](8.pdf "fig:")

![Relations between different energy conditions and the wormhole radius $r$. The green (long-dashed) line represents the expression $p+3\rho$, the blue (short-dashed) line represents the expression $p+\rho$, the pink (dot-dashed) line represents the expression $\rho-p$, and the red (solid) line represents the expression $\rho$.[]{data-label="f8"}](9.jpg)

Energy conditions
=================

In general, there are two important types of energy conditions in classical general relativity, i.e., pointwise energy conditions and averaged energy conditions. Generally, the standard pointwise energy conditions contain the above-mentioned NEC, the weak energy condition (WEC), the strong energy condition (SEC), and the dominant energy condition (DEC). The mathematical description of different energy conditions in a spatially flat Friedmann–Robertson–Walker (FRW) universe can be expressed as follows: $${\rm NEC} \Leftrightarrow \rho+p\geqslant0, \label{19}$$ $${\rm WEC} \Leftrightarrow \rho+p\geqslant0\quad {\rm and} \quad\rho\geqslant0, \label{20}$$ $${\rm SEC} \Leftrightarrow \rho+p\geqslant0\quad {\rm and} \quad\rho+3p\geqslant0, \label{21}$$ and $${\rm DEC} \Leftrightarrow \rho\pm p\geqslant0\quad {\rm and} \quad\rho\geqslant0. \label{22}$$ It is worth noticing that the NECs are the necessary conditions of the other three energy conditions mathematically.

In the present work, we are dedicated to 