Unruh effect of multiparticle states and black hole radiation

Figure  1.  Rindler space. Spacetime is divided into four wedges, left “L”, right “R”, future “F”, and past “P”. The black full line represents the trajectory with constant acceleration. The blue long dashed line is an example of the Cauchy surface. Both the left and right wedges are required to cover the entire space.

Figure  2.  Coefficient function $f_n(q, \tanh r)$ for different $q$. The horizontal axis represents $n$, and the vertical axis represents the value of $f_n(q, \tanh r)$. The figures show that as $q$ increases, the peak of $f_n(q, \tanh r)$ moves to the right, and the distribution becomes more “even”.

Figure  3.  Coefficient function $f_n(q, \tanh r)$ for different $\tanh r$ (different acceleration). The horizontal axis represents $n$, and the vertical axis represents the value of $f_n(q, \tanh r)$. The figures show that as $\tanh r$ (larger acceleration) increases, the peak of $f_n(q, \tanh r)$ moves to the right, and the distribution becomes more “even”.

Figure  4.  Right wedge von Neumann entropy for different $|q, 0\rangle_\alpha$ states along with $\tanh r$. Four curves from bottom to top are $q=0, 1, 10, 50.$ Entropy increases as acceleration increases. Simultaneously, a larger $q$ leads to larger entropy.

Figure  5.  “Normalized entropy” along with $\tanh r$ to compare right wedge entropy of different $q$ when fixing the energy. Four curves from bottom to top are $q=50 $, $10,\; 1,\; 0$. This shows that vacuum $q=0$ leads to the largest entropy, and a larger $q$ leads to less “normalized entropy”.

Figure  6.  Entropy of the radiation after $m$ steps when considering the simplification situation that assumes fixed surface gravity and $m\to \infty$. The vertical axis corresponds to the von Neumann entropy of radiation, and the horizontal axis is $\tanh {r_{m-1}}$. This quantifies the surface gravity of the last radiating process. The results show that the radiation entropy increases from zero at zero surface gravity and decreases to zero at infinitely large surface gravity.

Figure  7.  Entropy of the radiation after $m$ steps when considering surface gravity changes during the radiation process. In this case, we manually select certain values of $r_i$. The vertical axis is the von Neumann entropy of radiation, and the horizontal axis is $\tanh {r_{m-1}}$, which quantifies the surface gravity of the last radiating process. The results show that the radiation entropy increases from zero at zero surface gravity and decreases to zero at infinitely large surface gravity.