本篇論文中，我們在廣義相對論的範疇之內，研究奇異等溫球(一團球對稱的等溫氣體，具有自相似的性質，中心的密度為無限大，對應了一個奇異點)的重力塌縮現象。針對這個問題，我們試著以震波的存在，來延伸原本由中心向外塌縮的數學解。與僅考慮牛頓力學的情形不同，廣義相對論中，震波發生的原因，乃是奇異等溫球內部的重力塌縮而產生的時空變化；一旦氣體球的中心開始塌縮形成黑洞，時空的改變會導致塌縮區域之外的重力降低，因此產生一個由中心向外、超音速的震波。此一相對論性的震波與牛頓力學下的震波不同，並不需要額外的能量來維持震波的存在，而是時空本身變化的結果。我們認為此一震波是中心的黑洞釋放重力位能的一個現象。本篇論文中，震波的存在受到等溫氣體聲速的限制，我們計算的結果顯示，只有聲速的平方在 17.32% ~ 45.83% 光速的範圍之內，可以找到對應的震波解。在低聲速的情況下，此一問題的數學形式會逐漸趨近於牛頓力學的結果，由於我們所計算的震波僅存在廣義相對論的範疇之內，故在聲速過低的情形下，無法找到對應的震波解；而聲速的上限則可能與奇異等溫球本身的物理特性有關。對應到不同的聲速，我們所計算出的能量釋放比率約為中心黑洞靜止質量的3% ~ 15%，若假設中心的黑洞重十個太陽質量，所釋放的重力位能可達 爾格。此一龐大的能量若全以電磁波的形式釋放出來，與最強烈的伽瑪射線爆發所需的能量相當接近。

We studied the collapse solution to a singular isothermal sphere with consideration of relativistic collapse solution, and extended the collapse solutions with
critical points into the shock wave solution. Apart from its Newtonian counterpart, the general relativistic shock wave is generated by the change of space-time after the collapse of the sphere begins. The shock wave propagate outward at a super sonic speed, accompanied by the collapse of the sphere inside and the formation of a central black hole. In our calculation, the shock wave solutions are capped by the isothermal sound speed. The solution only exists for the sound speed ranged at 17.32% 45.83% of speed of light. We postulate this shock wave can release the gravitational binding energy of the central black hole by the ratio of 3% 15% of its rest mass. A simple estimation shows that the released energy is about 1054erg for a black hole with mass MBH ∼ 10M☉, which correspond to the energy scale of those most energetic gamma-ray bursts in the sky.

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