How much time does it take for the gravitons generated by a black hole singularity to travel before exerting gravity forces on other celestial bodies?

Question

It is known to all that the travelling speed of gravitons (the propagation speed of gravitational field) is not instant. So for black holes, the gravitons (the gravitational field) generated by the singularity of the black holes needs time to travel before exerting gravitational pulling forces on other celestial bodies. But,according to gravitational time dilation, the time near black holes are extremely dilated to infinity. This indicates, given the limited age of the universe, the gravitons (gravitational field) generated by the singularity of any black holes have NOT exert any pulling forces on any celestial bodies yet. Thus, we can conclude, it is impossible for any current celestial body to be pulled by the gravity of any black hole and orbit around a black hole. And, actually there should be no celestial bodies now gravitationally pulled by any black holes at all. But, this conclusion is apparently absurd. Any friends can explain this please?

Answer 1

It is known to all that the travelling speed of gravitons (the propagation speed of gravitational field) is not instant.

True. Changes to the gravitational field propagate at the (local) speed of light.

So for black holes, the gravitons (the gravitational field) generated by the singularity of the black holes needs time to travel before exerting gravitational pulling forces on other celestial bodies.

The gravitational field of a real, astrophysical black hole is not generated by the singularity. It is generated by the matter falling into the black hole when it formed, before it reaches the event horizon. (In the theoretical case of an eternal black hole, which has always existed in an infinite past, it is generated by the past singularity inside the white hole region, behind a second event horizon in the distant past. Our universe only goes back 14 billion years, so this isn't relevant for us, but I thought I'd mention it because you often see them discussed.)

The best way I've found to understand this is to draw the spacetime diagram of a black hole in Kruskal-Szekeres coordinates. This twists the coordinates round so that light travels at $45^\circ$ to the horizontal and vertical, local time runs up the page, and the radial in/out coordinate is horizontal. Distances and lengths are distorted. To see the conventional 'black hole sitting in spacetime' picture, tilt your head $45^\circ$ to the right. You can see the singularity at the centre of the black hole curving away to the left, the event horizon a straight line, and then the universe outside the black hole curving away to the right. The separation between them remains constant over history. The curvature of the lines is just because of the twisted-up coordinates we're using - in our effort to fit warped 4D spacetime on to a flat 2D page.

From any vantage point, you can only see things and be affected by things in your past light cone. So draw two lines at $45^\circ$ to the vertical down the page, and that is what you can see. It is also the source of the gravitational field where you are.

Armed with this picture, we can see that the singularity into which all the matter is falling is in the future of everyone. It can never be seen by anyone. It is not the cause or source of anything - the future cannot affect the past. The event horizon is in the future of everyone in the universe outside, and again cannot be seen any more than you can see or sense next week from where you are sitting.

If you trace back those $45^\circ$ lines from any point outside the event horizon, what you see is all the matter that has fallen into the black hole in the past, right back to the original collapsing star. This is the source of the gravity, and the source of the light you see when you look directly towards the black hole.

An observer hovering a fixed distance outside the black hole follows the hyperbolic curve seen on the right of the diagram. As the observer proceeds up to the top-right, the light they see is stretched out in time. The last few microseconds of light from the collapsing star are spread out over the millions of years the black hole exists, causing it to be extremely faint and extremely red-shifted. That is why a black hole appears black.

Although the matter of the collapsing star has long ago passed the event horizon and hit the singularity, the infinite time dilation on the event horizon means that the light and gravity it sourced are preserved in the space just outside the event horizon continue to 'leak out' forever. The gravity you experience at a distance was established by the star long before it collapsed - and is completely unchanged by the collapse process. There was one star's-worth of mass there long before the collapse, and there is still one star's-worth after. But the distant observer can only 'see' the gravity from the very last moments of the star as it fell screaming towards the event horizon, nearly frozen in time, and taking billions of years to escape.