# Can a sufficiently large black hole have low time dilation near its event horizon?

First time posting here, please let me know if I miss anything!

As I understand it, as the mass of a BH increases, its Event Horizon increases more than the rate of the mass being put in. In supermassive Black Holes, if the Event Horizon is far away from the BH mass, can there be a point at which gravitational time dilation has little effect while near the Event Horizon? Let's say near is a few tens of kilometers.

This question was inspired by the below questions:

As the mass of a Black Hole increases, the tidal forces present at its event horizon decreases. As we get to the super-massive black hole limit, the tidal forces may be so low that an observer might go through the event horizon unfazed.

The time-dilation between an observer hovering close by to the Event Horizon and an observer "at infinity"; however, will always approach infinity as the hovering observer moves closer and closer to the Event Horizon itself. Similarly, the redshift experienced by light as it escapes from nearby the horizon will approach infinite as the light is emitted closer and closer to the horizon itself. These effects will hold regardless of the mass of the black hole. In fact, these effects are related to the very definition of "event horizon" itself.

A non-spinning black hole without charge can be analyzed using the Schwarschild metric. The Schwarzschild radius (aka event horizon), measured as the circumference divided by $$2 \pi$$ is directly proportional to the mass, $$2GM/c^2$$ , where $$G$$ is the gravitational constant, $$M$$ is the mass of the black hole, and $$c$$ is the speed of light. So the area of the event horizon increases as the square of the mass, but the radius increases only proportionally to the mass.

The time dilation for a non-moving object measured by a distant observer, derived from the Schwarzschild metric would be

$$\left(1-\frac{2GM}{c^2r}\right)^{.5}\,,$$where $$r$$ is the radius (or distance) or $$\left(1-\frac{R_{\text{S}}}{r}\right)^{.5} \, ,$$ where $$R_{\text{S}}$$ is the Schwarzschild radius shown above.

So at the event horizon, the time dilation is infinite regardless of the mass (and therefore size) of the black hole. But if you take a fixed distance from the Event Horizon, you would have little time-dilation when $$r$$ is much larger than $$R_{\text{S}}$$. Given your numbers, the time dilation at $$r = 30 \, \mathrm{km}$$ would be .983 (1.000 would be no time-dilation) for a $$1 \, \mathrm{km}$$ $$R_{\text{S}}$$ black hole. But it would be much higher $$30 \, \mathrm{km}$$ past the event horizon of a supermassive black hole.

• @nat Thanks for the formatting. I'll look into how to do this. – Stuart Van Horne Sep 29 '18 at 15:27