Who Will Win the Race at the Event Horizon?

[arkantos]

Hi guys.

Imagine that in the exact instant when a massive particle A crosses the event horizon of a black hole, a Photon does the same,so that they have a race toward the singularity. Who will win the race? Will they have still different velocities?

 

[Orodruin]

Hi guys.

Imagine that in the exact instant when a massive particle A crosses the event horizon of a black hole, a Photon does the same,so that they have a race toward the singularity. Who will win the race? Will they have still different velocities?

You need to specify what you mean by "win the race". If they cross the horizon at the same space-time event, they will no longer be co-located inside the black hole. You therefore need to specify a simultaneity convention with which your "race" is being set up.

 

[Ibix]

Who will win the race? Will they have still different velocities?

As Orodruin notes, there are quite a lot of subtleties to answering your questions.

It's certainly the case that, if your massive particle has a little bit of length, the light will catch the rear of it and then pass the front of it. So yes, the two things do have different velocities in this sense.

"Who wins the race to the singularity" is hairier. The events where any two things strike the singularity are spacelike separated, so there is no invariant answer. At least one coordinate system (Kruskal-Szekeres interior Schwarzschild coordinates) assigns the same timelike coordinate everywhere on the singularity, so you are free to argue that all races to the singularity, no matter the contestants or any handicap, end in draws!

 

[kent davidge]

At least one coordinate system (Kruskal-Szekeres coordinates) assigns the same timelike coordinate everywhere on the singularity

Are you sure? Sorry if I seem to be arguing, this is just a question.

 

[Ibix]

Are you sure? Sorry if I seem to be arguing, this is just a question.

No - you're right. Interior Schwarzschild coordinates do what I meant - with that correction the point stands, I think. I'll correct it above.

Edit: changed my mind. Although interior Schwarzschild coordinates do assign the same time coordinate to the end of the race, they don't assign any coordinate at all to the start of the race, since they don't cover the event horizon. I've struck out that part of my last paragraph.

 

[PeterDonis]

they have a race toward the singularity. Who will win the race?

The singularity is a moment in time, not a place in space. So your question is like asking "Who will win the race to next Tuesday?"

 

[PAllen]

Another relevant, oft made point, is that the singularity is best thought of a missing spacelike line not a missing point. Thus, if the mental model was to have someone waiting at the finish, timing the race, the fundamental problem is that the photon and massive body cannot reach the same point on the singular line, that is, they cannot arrive at the same place, thus they can't have the same race.

Note that if non-free fall paths are allowed, two material test bodies can race to the same place on the singularity. But then, you still can't declare a winner, because this (limiting) event has a time as well as place, so they have arrived at the same time by construction.

 

[PAllen]

A different, possibly amusing, form of race is possible for two test bodies crossing the horizon together, then racing on any time like path they want to anywhere on the singularity - the winner being declared as the one with the smallest time on their watch starting from the mutual crossing. What's funny about it is that the winner can approach zero time passed to the singularity following a near lightlike path.

 

[arkantos]

Thank you for your answers.

Sorry if this will seem silly: At least we can say that the two particles that enter in the same event horizon at the same time toward the same singularity will have the same speed of light velocity; it is correct? And this can be said for every particle that enters in a given event horizon. This is the point that I wanted to ascertain.
I am more interested in the philosophical perspective.
If special relativity implies an infinite amount of energy to reach speed of light for a massive particle, and this is what seems to happen in a black hole, why Physicists consider the concept of singularity a mathematical aberration?, a symptom that GR is incomplete?
I mean, if what I said is correct, why the idea of singularity is more absurd than the idea of a massive particles that travel toward the singularity at the speed of light?

 

[Vanadium 50]

First, we don't discuss philosophy here.

Second, the physics in your message is incorrect, so we can't use it as a sensible basis for discussion.

 

[Ibix]

At least we can say that the two particles that enter in the same event horizon at the same time toward the same singularity will have the same speed of light velocity; it is correct?

No. That was a point I made in post #3.

 

[PeterDonis]

At least we can say that the two particles that enter in the same event horizon at the same time toward the same singularity will have the same speed of light velocity; it is correct?

No. As @Ibix said, this was already explained.

I am more interested in the philosophical perspective.

As @Vanadium 50 has pointed out, this is not a philosophy forum, it's a physics forum. Even leaving that aside, you can't have valid philosophy if you have an incorrect understanding of the physics.

If special relativity implies an infinite amount of energy to reach speed of light for a massive particle, and this is what seems to happen in a black hole

It isn't. When a massive particle crosses the hole's horizon, the horizon is moving outward at the speed of light.

You might want to read this Insights article series on the Schwarzschild geometry:

https://www.physicsforums.com/insights/schwarzschild-geometry-part-1/

It addresses a number of common misconceptions about black holes, some of which you seem to have.

if what I said is correct

It isn't. This has been explained.

 

[PeterDonis]

The OP question has been answered. Thread closed.