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If black holes swallow everything and assumes that the black holes do not get out the things from them, then can we consider a black hole as an infinite delay system? It takes the input and it is a super slow system that has an infinite delay so they do not give an output?

If yes, How can model such a system? I thought about modeling the system like this:

y(x) = x + jz.

where x is the input and j is the imaginary part. z is a parameter within the system(black hole) itself.

the phase shift for such a system can be given by:

tan-1(z/x) and this quantity must approaches to infinity which is impossible since the maximum value for the function is pi/2. So how can we adjust the parameter z to give the infinite delay?

A side note: I gave the black hole as an example, so the question is mainly about how to model the infinite delay system.

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  • $\begingroup$ Maybe as $y(t)=0$ ? $\endgroup$
    – MBaz
    Jan 22 '17 at 22:27
  • $\begingroup$ @MBaz sorry the parameter is z not y. So, the parameter z has nothing to do with output. Yes, y(t) = 0. However, I want the system to have an infinite delay, but not zero output (I don't know if this a valid argument) $\endgroup$
    – hbak
    Jan 22 '17 at 22:33
  • $\begingroup$ So what is the output for $t<\infty$? $\endgroup$
    – MBaz
    Jan 23 '17 at 3:25
  • $\begingroup$ @MBaz based on my question it's zero. $\endgroup$
    – hbak
    Jan 23 '17 at 4:30
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    $\begingroup$ $y(t)=x(t-5000)$ (where time is measured in years). Note that a factor $\text{exp}(j\phi)$ is equivalent to a time delay only for sine and cosine signals. $\endgroup$
    – MBaz
    Jan 23 '17 at 15:11
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Infinite delay means nothing happen. You can assume uncontrollability or you can assume a system with zero dynamics and zero gain and some arbitrary initial condition.

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