Black Holes with Paul M. Sutter

The mrgirl Podcast

Notes on disagreements:

Infinite time dilation near an event horizon:

At around 15:30, Paul and I disagreed about what an observer falling into a black hole would experience in terms of time dilation.

Paul proposed a 2X limit on how quickly clocks would run in this observer's frame, but this limit only applies in a very specific scenario, and I believe we got our wires crossed here.

In the standard framework of general relativity, it's important to distinguish between what is occurring in an observer's frame versus what they are physically perceiving. For example, there may be a galaxy merger occurring right now that we will not perceive for millions of years.

When Paul referenced a 2X limit on how quickly external clocks would run in that infalling observer's frame, he was referring to the maximum speed of pulses the observer would physically receive from a clock placed at infinity (in flat Minkowski spacetime). For this limit to apply, the observer also has to have fallen from rest at infinity, and it only applies to photons arriving from directly behind the observer. If the clock is placed at any other angle or the observer starts their fall from less than infinity, the observer will receive signals arbitrarily fast.

(The 2x limit in this very specific scenario is the result of a competing redshift and blueshift. The velocity of the infalling observer relative to the clock approaches light speed, causing a Doppler redshift: the frequency of pulses they receive slows down because the observer is nearly outrunning them. At the same time, the observer is approaching a black hole, so gravitational time dilation increases toward infinity, causing a gravitational blueshift: the frequency of received light pulses increases. Both effects mediate each other such that the maximum rate of pulses received is two ticks per second on the infalling observer’s clock.)

If the observer starts their fall from less than infinite distance, or if the clock is closer than infinite distance, or if the observer accelerates or decelerates, the 2X balance will be thrown off.

For any angle other than directly behind the observer, the blueshift increases without bound and the frequency of received pulses will approach infinity.

This means the the 2X limit is actually a blueshift floor, not a ceiling. Even if you were to fly a spaceship at top speed into a black hole, photons arriving directly behind would end up at a 2X frequency compared to your watch. All other angles would arbitrarily increase toward infinity.

As I mentioned, the frequency of pulses received by the observer is different from “what is currently happening” in the observer's frame (in the standard framework). As an infalling observer approaches an event horizon, all clocks in the universe will indeed tick arbitrarily fast in the observer’s frame, regardless of what signals they receive.

Our disagreement about whether time dilation can be interpreted equally and oppositely from either perspective seems to have arisen from this confusion about whether we were talking about “what is happening” in an observer frame versus what those observers would physically perceive in terms of received light signals.

This whole argument is a bit confusing because I usually argue against the standard definition of “what is happening” and actually prefer the observer’s received signals as the definition of reality. But since the argument is about how time dilation works in the framework of general relativity, I want to be careful and clear about differentiating these concepts.

I also don't believe that black holes can physically exist as fully formed structures identical to the mathematical objects, because they would take infinite time to form (this was Einstein's view).

Can you physically see the curve of a light path originating from your location?

Paul hypothesized that if you shined a laser near a massive object, you could use dust to scatter the light rays and thus visually reveal spacetime curvature.

This would not work because the scattered rays would follow the same curvature back to your eye, appearing to follow straight lines from your perspective. You would need to move to a different vantage point to see the curved path of the laser through the dust.

Can you detect the location of a singularity if you're in a black hole?

No, and for the same reason you can't visually see the laser curve from its origin point.

In a black hole, there is no other location from which you can see curvature towards the singularity: you are always at the origin.

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