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codebpr
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- TL;DR Summary
- I want to plot Hawking temperature as a function of z(inverse of horizon radius), which requires making use of a form factor A(z). I want to make an ansatz for form factor such that I get the desired plots.
This is basically a physics problem but I will try my best to highlight the mathematics behind it.
Suppose I have two functions:
$$T(z,B)=\frac{\text{z}^3 e^{-3 A(\text{z})-B^2 \text{z}^2}}{4 \pi \int_0^{\text{z}} \xi ^3 e^{-3 A(\xi )-B^2 \xi ^2} \, d\xi },$$
$$\phi(z,B)=\int_0^z \sqrt{-\frac{2 \left(3 x A''(x)-3 x A'(x)^2+6 A'(x)+2 B^4 x^3+2 B^2 x\right)}{x}} \, dx$$ where [itex]z \in \mathbb{R^+} [/itex] and [itex]B \in [0,1][/itex]
and I want to find a function [itex]A(z)[/itex], which is known as the form factor in literature, such that the plot of the function [itex]T(z,B)[/itex] v/s [itex]z[/itex] has one local minimum along with the condition that [itex]T(z,B)\rightarrow\infty[/itex] when [itex]z\rightarrow0[/itex], also [itex]\phi(z)[/itex] is real-valued. When I take the ansatz [itex]A(z)=-a z^2[/itex], I am able to satisfy the above condition for [itex]B\in[0,0.6][/itex] and get plots like:
Now for a different model, I need to use such an ansatz for [itex] A(z) [/itex] such that I may be able to satisfy the real valued-ness of [itex] \phi(z) [/itex] and get a local minimum as well as a maximum for the plot of [itex] T(z,B) [/itex] v/s [itex] z [/itex] with the condition that [itex] T(z,B)\rightarrow\infty [/itex] when [itex]z\rightarrow0[/itex] and [itex]T(z,B)\rightarrow0[/itex] when [itex]z\rightarrow\infty[/itex] to get plots like these:
If possible I want to keep [itex]B\in[0,1][/itex]. The constant [itex]a[/itex] has to be used in the form factor somehow, whose value is 0.15. The form factor can also be written in terms of [itex]A(z,B)[/itex]. Is there a way to use Mathematical analysis to come up with such a form factor? Any help in this regard would be truly beneficial!
Suppose I have two functions:
$$T(z,B)=\frac{\text{z}^3 e^{-3 A(\text{z})-B^2 \text{z}^2}}{4 \pi \int_0^{\text{z}} \xi ^3 e^{-3 A(\xi )-B^2 \xi ^2} \, d\xi },$$
$$\phi(z,B)=\int_0^z \sqrt{-\frac{2 \left(3 x A''(x)-3 x A'(x)^2+6 A'(x)+2 B^4 x^3+2 B^2 x\right)}{x}} \, dx$$ where [itex]z \in \mathbb{R^+} [/itex] and [itex]B \in [0,1][/itex]
and I want to find a function [itex]A(z)[/itex], which is known as the form factor in literature, such that the plot of the function [itex]T(z,B)[/itex] v/s [itex]z[/itex] has one local minimum along with the condition that [itex]T(z,B)\rightarrow\infty[/itex] when [itex]z\rightarrow0[/itex], also [itex]\phi(z)[/itex] is real-valued. When I take the ansatz [itex]A(z)=-a z^2[/itex], I am able to satisfy the above condition for [itex]B\in[0,0.6][/itex] and get plots like:
Now for a different model, I need to use such an ansatz for [itex] A(z) [/itex] such that I may be able to satisfy the real valued-ness of [itex] \phi(z) [/itex] and get a local minimum as well as a maximum for the plot of [itex] T(z,B) [/itex] v/s [itex] z [/itex] with the condition that [itex] T(z,B)\rightarrow\infty [/itex] when [itex]z\rightarrow0[/itex] and [itex]T(z,B)\rightarrow0[/itex] when [itex]z\rightarrow\infty[/itex] to get plots like these:
If possible I want to keep [itex]B\in[0,1][/itex]. The constant [itex]a[/itex] has to be used in the form factor somehow, whose value is 0.15. The form factor can also be written in terms of [itex]A(z,B)[/itex]. Is there a way to use Mathematical analysis to come up with such a form factor? Any help in this regard would be truly beneficial!
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