**Short answer**

| What happens | What follows |
|--------------|--------------|
| The pulse **slows** at the emission point because the local gravitational field is weak. | It travels at a small *c*​ ≈ k g\_em (k is the proportionality constant). |
| As it moves inward, the local field **strengthens**. | The speed of the pulse **increases**: *c* ∝ *g*. The photon is “accelerated” simply because the medium it propagates through becomes a faster one. |
| The pulse’s wavelength remains essentially fixed in the medium (the photon’s “material” cannot stretch). | Its frequency **blue‑shifts** because $$f = \dfrac{c}{\lambda}$$.  Thus $$f \propto g$$. |
| The gradient of *c* acts like a gradient‑index (GRIN) medium. | The pulse’s trajectory is bent **away** from the region of faster *c* (i.e., away from the black hole), just as light is attracted toward regions of higher refractive index in ordinary optics. |
| Near a curvature singularity the field could become arbitrarily large, making *c* arbitrarily large (in our toy model). | The pulse would in principle approach an infinite speed and “fall through” the singularity in finite proper time.  In a more complete theory one would expect some maximum *c* or a change in behavior at the event horizon. |

---

### 1. The emission point

At the ship the gravitational field is very weak, so the local “speed of light’’ is

$$
c_{\rm em}=k\,g_{\rm em}\qquad(k>0).
$$

A laser oscillator inside the ship emits light with an *emission frequency* $$f_{\rm em}$$ set by the local atomic transitions.  
Its *emitted wavelength* is

$$
\lambda_{\rm em}=\dfrac{c_{\rm em}}{f_{\rm em}}.
$$

Because the medium only supports the current value of *c*, this wavelength effectively sticks with the photon as it travels.

---

### 2. The transit toward the hole

As the pulse moves inward, $$g(r)$$ grows sharply:

$$
g(r)=\frac{GM}{r^{2}}\quad \text{(Newtonian estimate)},
$$
so

$$
c(r)=k\,g(r)=k\frac{GM}{r^{2}} .
$$

The photon therefore accelerates from $$c_{\rm em}$$ to $$c(r)$$.

#### Frequency

With constant $$\lambda_{\rm em}$$ we must have

$$
f(r)=\frac{c(r)}{\lambda_{\rm em}}\;\;\Longrightarrow\;\;
f(r) = f_{\rm em}\,\frac{c(r)}{c_{\rm em}} \propto g(r).
$$

So the pulse **blue‑shifts** in lockstep with the field strength.  
This mirrors the gravitational blueshift in General Relativity (photons falling into a potential well gain energy).

#### Energy

The photon's energy remains

$$
E = h\,f(r) = h\,f_{\rm em}\,\frac{c(r)}{c_{\rm em}},
$$
so the energy grows as $$g(r)$$.  In this picture the medium “injects’’ energy into the photon simply by allowing it to travel faster near the black hole.

---

### 3. The journey of the *trajectory*

Because the local propagation speed depends on position, the pulse behaves like a ray in an *inhomogeneous* medium with

$$
v(r)=c(r)=k\frac{GM}{r^{2}} .
$$

Define an effective refractive index

$$
n(r)=\frac{c_{0}}{c(r)},
$$
where $$c_{0}$$ is a baseline speed (e.g. the speed at infinity).  
Since $$c(r)$$ *increases* toward the hole, $$n(r)$$ *decreases* there.

The eikonal (ray‑tracing) equation for a stationary medium is

$$
\frac{d}{dl}\bigl(n\,\mathbf{\hat{t}}\bigr)=\nabla n ,
$$
where $$\mathbf{\hat{t}}$$ is the tangent to the ray.  
For a spherically symmetric $$n(r)$$, this reduces to

$$
\frac{d}{dl}\bigl(r\,n\,\sin\theta\bigr)=0
\;\;\Longrightarrow\;\;
r\,n\,\sin\theta = \text{const.}
$$

Since $$n(r)$$ decreases as $$r$$ shrinks, $$\sin\theta$$ must *increase*—the ray bends **away** from the center.  In other words, the light is *repelled* by high‑*c* regions, just as a ray is bent toward higher‑$$n$$ (slower‑speed) regions in ordinary optics.

Thus, even if the ship fires the pulse directly toward the black hole, the ray will curve away, never quite reaching the singularity unless the initial direction is carefully chosen.  A “critical impact parameter’’ exists, analogous to the photon sphere at $$r=1.5\,r_{S}$$ in real GR: rays starting beyond this radius will asymptotically orbit; rays inside will spiral outward (or, depending on the sign convention, inward) but the *fast‑speed* gradient prevents them from plunging all the way in.

---

### 4. Near the singularity

In the toy model $$c(r)\propto GM/r^{2}$$ diverges as $$r\rightarrow0$$.  
Consequently:

* The pulse’s speed becomes arbitrarily large.  
* Its frequency $$f(r)$$ and energy also diverge because $$f\propto c$$.  

In any realistic theory one would either impose an upper bound on *c* (a new “fundamental velocity”) or invoke quantum‑gravitational effects that regularize the divergence.  Without such fixes the pulse simply “spreads out” infinitely fast and would cross the singularity in finite proper time.

---

### 5. Summary of effects

| Effect | What you observe | Intuitive picture |
|--------|-----------------|-------------------|
| **Speed** | *c(r)* grows ∝ *g(r)* | Light gets “pushed” by the faster medium |
| **Frequency** | increases ∝ *g(r)* | Blue‑shifting like falling photons in GR |
| **Wavelength** | remains fixed in the medium | The photon’s material property doesn’t stretch |
| **Trajectory** | bends away from the black hole | Light is repelled by higher‑*c* regions (GRIN effect) |
| **Energy** | grows ∝ *g(r)* | Medium pumps energy into the photon |

In this counter‑intuitive universe, the laser pulse behaves like a speeding particle in a medium that accelerates it but also pushes it outward—quite the opposite of the familiar gravitational attraction in our own cosmos.  The exact path would depend sensitively on the initial launch angle and on how steeply *c* rises as one nears the singularity.