0% of speed of clock outside infinite well

Question

Juan Maldacena says here: "For a very massive and very compact object the deformation (or warping) of spacetime can have a big effect. For example, on the surface of a neutron star a clock runs slower, at 70 percent of the speed of a clock far away. In fact, you can have an object that is so massive that time comes to a complete standstill. These are black holes."

(from https://www.ias.edu/ideas/2011/maldacena-black-holes-string-theory)

It is very clear that he refers to time-dilation and that the clock runs at 70% of the speed of a clock in another reference frame. When the clock is at the event horizon though it stops. Obviously there is an infinity there. We can talk about relative speeds until we reach a 0%. "time comes to a complete standstill"

When I ask about black holes many people point to the central singularity. It seems though that for time to come to a halt at the event horizon we must also call this a singularity. Where am I wrong in my thinking?

For time to stop there at EH curvature must be already at maximum (if a maximum is possible). How can we proceed to think beyond this and why do so many people ignore the logical singularity at the horizon in favour of a central one?

I am not asking for opinions - what I mean specifically is why we still assume there is a central singularity when the metric arrives at zero long before that - why is that approach still taken?

and can a zero be not absolute when one considers quantum mechanics?

Answer 1

It seems though that for time to come to a halt at the event horizon we must also call this a singularity.

Gravitational time dilation depends on the gravitational potential. A singularity (the type of singularity we mean when we talk about singularities in GR) is a certain type of misbehavior of the curvature, which causes geodesic incompleteness. Curvature and gravitational potential are two different things.

Any function can have a singularity, but this mathematical definition of a singularity is different from the very specialized usage in GR.

Answer 2

The word "singularity" is already widely used in mathematics, before one looks at any examples in physics, so one has to be careful.

In the case of the spacetime at and around a black hole, there are two types of observation in which the word "singularity" is commonly used. In one observation, we adopt a certain coordinate system, called Schwarzschild coordinates, to map spacetime, and we find extreme behaviour at the horizon. But it turns out that the curvature of spacetime is not infinite at the horizon, whereas the curvature does tend to infinity at another location in spacetime, the location which is assigned the value $r=0$ in Schwarzschild coordinates. So the standard terminology is to say that a curvature singularity occurs at $r=0$ and a coordinate singularity occurs at $r=r_s$ in Schwarzschild coordinates. But one does not have to map spacetime using Schwarzschild coordinates. In anther coordinate system there need not be any coordinate singularity at the horizon, but there will always be a curvature singularity at the location of the curvature singularity, because curvature is absolute; it is a property of spacetime itself, not of coordinates.

These statements can be used to justify various physical predictions. One prediction is that if one were to lower a clock on a rope to a place just outside the horizon, and then raise the clock, then the total time elapsed on the clock will be much less than the time elapsed on a similar clock residing permanently at some high place well away from the horizon. Furthermore, the fraction of the overall evolution of the lowered and raised clock which takes place while it resides near the horizon tends to zero as that location approaches the horizon. Other types of experiment can also be used to confirm this. One can imagine that the clock stays near the horizon and sends out light signals, for example. So you are right to say that a certain kind of mathematical singularity is associated with observable physical effects connected to the horizon.

Nevertheless, the standard terminology in which we simply say "the singularity" when referring to the curvature singularity is a good terminology. It is a shorthand for "the curvature singularity".

Finally, you are right to suspect that quantum physics implies that statements about horizons and singularities made by classical general relativity do not capture all the relevant physics. In particular, black hole entropy and Hawking radiation is connected to this.

Answer 3

Singularities mean the value becomes infinite .( For example it happens for 1/r potentials when r=0.) You could call it a discontinuity. The metric is not zero at the black hole singularity, but infinite.

A gravitational singularity, spacetime singularity or simply singularity is a location in spacetime where the gravitational field of a celestial body is predicted to become infinite by general relativity in a way that does not depend on the coordinate system.

Italics mine.

The Schwarzschild metric has a singularity for $r=0$ which is an intrinsic curvature singularity. It also seems to have a singularity on the event horizon $ r=r{_{s}}$. Depending on the point of view, the metric is therefore defined only on the exterior region $ r>r{_ {s}}$, only on the interior region $ r<r{_{s}}$ or their disjoint union. However, the metric is actually non singular across the event horizon as one sees in suitable coordinates (see below).

One has to consider the metric, as shown in the link. There is though a discontinuity.

There is also this observation:

Solutions to the equations of general relativity or another theory of gravity (such as supergravity) often result in encountering points where the metric blows up to infinity. However, many of these points are completely regular, and the infinities are merely a result of using an inappropriate coordinate system at this point.

So one has to be wary of the mathematics, and which metric and what coordinate systems the statement "In fact, you can have an object that is so massive that time comes to a complete standstill" is made.

and can a zero be not absolute when one considers quantum mechanics?

Quantum mechanics introduces the Heisenberg uncertainty to all measurement results, sure.

Look how the Big Bang singularity is changed by the introduction of effective quantum mechanics at the singularity.