How would time dilation play into this? That observer would see the SMBH as further away at the time of the event, but they’d experience time at a slower rate. Would the extra time ever ‘beat’ the extra distance?

The observer moving away from the SMBH would see the event proceed more slowly. The light curve would therefore be stretched out in time --- the source would get brighter more slowly and would get dimmer more slowly. This is actually an important effect that needs to be accounted for in supernova observations in distant galaxies since time dilation is non-negligible.

Thanks, but I was actually wondering more about the apparent time of the flare (say, peak brightness since as you say the duration changes) based on the observer's speed.

Say we're both on Earth 26,000 years ago. You stay on Earth and I shoot off at C/2 in a direction away from the SMBH. You wait 26,000 years and then see the flare. How long do I wait in my own reference frame before I see the flare?

My guess intuitively is that time dilation and the extra distance cancel out and I still wait exactly 26,000 years (no matter how fast I go), but I'd have to dig out my old physics textbook to figure it out properly. Also, what happens if instead I shoot off towards the SMBH? Or at 90 degrees? I'd obviously see the flare sooner in my reference frame, but how much?

It seems like there's a general rule that, by moving, you can decrease the time until you see events (at least those communicated by light) but you can never increase it.

The Lorentz factor here is 1/sqrt(3/4), around 15% over unity. So the elapsed time for you to see the flare is 2/1.15 * 26ky, or 45 ky. In general, for any λ multiple of c, the elapsed time multiple is sqrt(1 - λ^2) / (1 - λ), which reduces to 1 + λ in the λ << 1 case. You can try plotting it for λ from -1 (c directly towards the event) to +1 (c directly away from the event) to get a feel for how it behaves.

That's great, thanks! For anyone else who wanted to see the graph, here's mine: https://www.desmos.com/calculator/kihnh5ohrv

If you (somehow) keep accelerating in one direction, there are events you will never observe. It's called a Rindler horizon. Your world-line traces a hyperbola... which will always outpace photons from certain spacetime locations.

https://youtu.be/7cj6oiFDEXc

I believe that because the speed of light is the same in any inertial frame reference.. If you were two light minutes away from the event in the SMBHs inertial frame at the event and traveling toward the earth at a speed sufficient to reach it at the same time as the light from the event.. You would arrive at the earth at the same time as the light but only 2 minutes will have passed in your frame of reference?

> but only 2 minutes will have passed in your frame of reference?

Only 2 minutes would have passed for the SMBH in your frame of reference (but 52k years would have passed for earth in your frame of reference)

from the black hole's perspective, the observer would be travelling slowly in time. But from the observer's perspective, the black hole would be travelling slowly in time (since the observer sees themselves as being at rest, and the black hole as shooting away from them).

But if you existed long ago wouldn't have time moved slower since you would have been closer to the SMBH and everything else (i.e., universe would be more dense). So according to general relativity wouldn't it also be right to say it happened less than 26,000 years ago from that reference frame?

Thank you. Make sense with a bit more of my thought!