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I have the following problem:

Suppose a test particle is radially freely infalling towards the Schwarzchild radius (r=2M). Suppose the particle is sending out a monochromatic electromagnetic signal out to $r=\infty$. The observer at infinity measures a redshifted wavelength $\lambda_{\infty}$ such that $$\lambda_{\text{particle}} \sim \lambda_\infty e^{-t/k}$$ where $k$ is a constant when the test particle is very close to crossing the horizon. Express $k$ in terms of the mass of the black hole.

Attempt at a solution: The frequency redshift for the Schwarzchild metric is $$\frac{\omega_1}{\omega_2} = \frac{(1-2M/r_2)^{1/2}}{(1-2M/r_1)^{1/2}} = \frac{\lambda_2}{\lambda_1}$$

Where $M$ is not the proper mass. I'll be taking $r_1 \to \infty,$ so $$\lambda_\infty (1-2M/r_2)^{1/2} = \lambda_{\text{particle}}$$ I need this in terms of $t$ so I look at a null geodesic: $$0=-(1-2M/r)dt^2 + (1-2M/r)^{-1}dr^2$$ $$(dt/dr)^2 = (1-2M/r)^{-2}$$ $$\Delta t = \pm \int_{r_1}^{r_2} (1-2M/r)^{-1}dr = \pm (2M\ln(r-2M)+r)\Big|_{r_1}^{r_2}$$ $$\Delta t =(r_1 -r_2) + 2M\ln \frac{r_1 -2M}{r_2 - 2M}$$

I'm not really sure how to take it from here. I don't understand how to change the expression involving $r_1,$ and $r_2$ into one involving $t.$ Any help would be appreciated.

Jbag1212
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    Your expression for the redshift is only true for static observers, whereas in this case you have a freefalling object emitting the signal. – ProfRob Apr 05 '21 at 22:36
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    @JohnRennie why do you think that is a duplicate? It doesn't even deal with the Doppler shift. For the record, I voted to close this as "homework". – ProfRob Apr 06 '21 at 09:37
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    See https://physics.stackexchange.com/questions/213441/do-free-falling-observers-see-gravitational-blueshift – ProfRob Apr 06 '21 at 16:41

1 Answers1

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The problem can be solved by deriving a correct expression for $\lambda_{\infty}(r)$. Then by differentiating with respect to $t$ we get $$\frac{d\lambda_\infty}{dt}=\frac{d\lambda_\infty}{dr}\frac{dr}{dt}.$$

If the right hand side is $\propto \lambda_{\infty}$ as $r \rightarrow r_s$, then I think you have your answer.

ProfRob
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