# 能量條件

## 數學陳述

• 弱能量條件(weak energy condition)規定對於每個指向未來(future-pointing)的「類時向量場」(timelike vector field) ${\displaystyle {\vec {X}}}$ ，相應的觀察者所觀察到的物質密度永不為負值：
${\displaystyle \mu =T_{ab}\,X^{a}\,X^{b}\geq 0}$
• 零能量條件(null energy condition)規定對於每個指向未來的「零向量場」(null vector field) ${\displaystyle {\vec {k}}}$
${\displaystyle \mu =T_{ab}\,k^{a}\,k^{b}\geq 0}$
• 強能量條件(strong energy condition)規定對於每個指向未來的「類時向量場」 ${\displaystyle {\vec {X}}}$ ，由相應的觀察者所測量到的潮汐張量(tidal tensor)的跡數(trace)永不為負值：
${\displaystyle \left(T_{ab}-{\frac {1}{2}}\,T\,g_{ab}\right)\,X^{a}\,X^{b}\geq 0}$
• 主能量條件(dominant energy condition)規定，除了弱能量條件成立之外，對於每個指向未來的「因果性向量場」(causal vector field)（也就是類時或零）${\displaystyle {\vec {Y}}}$ ，向量場${\displaystyle -{T^{a}}_{b}\,Y^{b}}$ 必須是指向未來的「因果性向量場」。換句話說，質能永遠不會被觀測到以超過光速的速度流動。

${\displaystyle \int _{C}T_{ab}\,k^{a}\,k^{b}\,d\lambda \geq 0}$

## 完美流體

${\displaystyle T^{ab}=\mu \,u^{a}\,u^{b}+p\,h^{ab}}$

${\displaystyle T^{{\hat {a}}{\hat {b}}}=\left[{\begin{matrix}\mu &0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{matrix}}\right]}$

• 弱能量條件規定${\displaystyle \mu \geq 0,\;\;\mu +p\geq 0}$
• 零能量條件規定${\displaystyle \mu +p\geq 0}$
• 強能量條件規定${\displaystyle \mu +p\geq 0,\;\;\mu +3p\geq 0}$
• 主能量條件規定${\displaystyle \mu \geq |p|}$

## 參考文獻

• Hawking, Stephen; and Ellis, G. F. R. The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press. 1973. ISBN 978-0-521-09906-6. The energy conditions are discussed in §4.3.
• Poisson, Eric. A Relativist's Toolkit: The Mathematics of Black Hole Mechanics. Cambridge: Cambridge University Press. 2004. ISBN 978-0-521-83091-1. Various energy conditions (including all of those mentioned above) are discussed in Section 2.1.
• Carroll, Sean M. Spacetime and Geometry: An Introduction to General Relativity. San Francisco: Addison-Wesley. 2004. ISBN 978-0-8053-8732-2. Various energy conditions are discussed in Section 4.6.
• Wald, Robert M. General Relativity. Chicago: University of Chicago Press. 1984. ISBN 978-0-226-87033-5. Common energy conditions are discussed in Section 9.2.