general relativity - What does black hole formation and evaporation actually look like as viewed from far away?

Many people on Physics SE (myself included) have been confused about what black hole formation and evaporation look like when viewed from far away. For example:

Does any particle ever reach any singularity inside the black hole?

Can black holes form in a finite amount of time?

How can anything ever fall into a black hole as seen from an outside observer?

From where (in space-time) does Hawking radiation originate?

How does the evaporation of a black hole look for a distant observer?

and numerous duplicates.

This confusion all arises from the same place: the apparent contradiction between (a) the fact that a Schwarzchild event horizon lies in a faraway observer's infinite future, so from her perspective infalling matter takes an infinite amount of time to actually hit the black hole, and (b) the following naive chronological sequence:

1. Black hole forms


2. More matter falls into black hole


3. Black hole grows


4. Black hole's Hawking radiation rate surpasses rate of matter infall


5. Black hole shrinks


6. Black hole evaporates completely

The unifying question is, how can a faraway observer observe all six of these developments in sequence, when the first one takes place in her infinite future?

In principle, this question can be mostly or completely answered by studying this Penrose diagram that I stole from Nathaniel:

Unfortunately, my intuitive understanding of Penrose diagrams isn't great, what with their mixing up and rescaling of the asymptotic space and time coordinates. I think that for a situation this subtle, a picture or video is worth a thousand words. Therefore, I (and I think many other SE users) would be eternally grateful, and I think many misconceptions would be cleared up, if someone could create a simple qualitative video visualization which translates this Penrose diagram into the perspective of a faraway observer (specifically, an observer following a timelike geodesic at such large asymptotic Schwarzchild $r$ coordinate that this $r$ coordinate changes negligibly over her perspective of the black hole's entire lifetime).

I'm thinking it would look something like this: a bunch of blue dots (representing matter) initially drifting inward, appearing to get "stuck" just outside of a growing black circle representing the absolute and/or apparent horizon, then yellow dots/rays representing Hawking radiation moving outward from just outside the horizon and passing the blue does, the black circle and its surrounding "layer" of blue dots shrinking, and the blue dots and black circle all finally collapsing to a point and sending out a big shower of yellow dots/rays. However, please correct any details of my mental picture that are incorrect (and feel free to change the suggested color scheme :-) ). Please feel to speed up or slow down the time scales for visual clarity (since I suspect a realistic black hole would just sit around for a few eons and then appear to vanish in a comparative instant), but please preserve the chronological ordering of observed events. Also, assume that the observer can detect arbitrarily low-frequency radiation, to overcome the huge redshift from matter very close to the apparent horizon. Thanks!

(PS To forestall the inevitable comments: yes, I'm fully aware that from the perspective of an observer falling into a black hole, crossing the horizon only takes finite proper time, and that he does not locally observe anything unusual as he does so.)

Edit: To clarify, I'm certainly not asking anyone to do a simulation of black hole collapse under realistic assumptions about the stress-tensor, or anything like that - just to take the the Penrose diagram above and figure out the conformal mapping that turns it back into the ordinary Schwarzchild coordinates. Not necessarily even mathematically precisely, just qualitatively would work.

general relativity - Why didn't Newton have a cosmological constant

Einstein initially added the Cosmological Constant because (if I get this right) it seemed to him that the universe should be static. I agree that back then this would have been an obvious assumption. I'm curious now, before Hubble, where there any opinions/debates about whether the universe would be expanding or contracting?

quantum field theory - Where does this term "shell" with prefix "on-/off-" come from?

Is there some historical reasons or is there a specific reason behind it?

This question is connected to: Why on-shell vs. off-shell matters?

Answer

A particle is said to be on-shell if it satisfies the relativistic dispersion relation,

$$E^2 = p^2 +m^2$$

in units wherein $c=\hbar=1$. If you graph it, you obtain a parabolic surface for massive particles, and a cone for massless particles, like a photon. This is known as the mass shell, it is quite literally a shell or surface. The momentum of a real particle can be represented as a vector on the surface, hence the expression on shell. Virtual particles do not have these on the surface, hence they are off shell.

---

Source: Perimeter Institute, A Deeper Dive: On Shell and Off Shell

everyday life - How do candles and wicks work?

The wick of my tea candle was buried in wax. So I lit a piece of paper and stuck it in the wax. Now the wax is burning off the paper, as if that were the wick.

The wax itself wouldn't light on fire without a wick, yet clearly what's burning is the wax, since the paper would have long ago burned up. What's going in here?

general relativity - Covariant derivative and Leibniz rule

I read the Wikipedia page about the covariant derivative, my main problem is in this part:

http://en.wikipedia.org/wiki/Covariant_derivative#Coordinate_description

Some of the formulas seem to lead to contradictions, I assume I'm making some mistakes.

Here are some formulas from that page.

They define the Covariant derivative in the direction $\mathbf e_j$, denoted $\nabla_{\mathbf e_j}$ or $\nabla_j$ so that:

$\nabla_{\mathbf e_j} \mathbf e _i = \nabla_j \mathbf e _i = \Gamma^k_{\ \ ij}\mathbf e_k$

And define it so it obeys Leibniz' rule.

They then go on to show that

Where it seems they used

$\nabla_{\mathbf e_i} u^j = \frac {\partial u^j}{\partial x^i}$

But then later they define here: http://en.wikipedia.org/wiki/Covariant_derivative#Notation

$\nabla_{\mathbf e_i} u^j = \frac {\partial u^j}{\partial x^i} + u^k \Gamma^j_{\ \ ki}$

1) Is this a misunderstanding of mine or a problem in Wikipedia?

Also instead of the definition:

$\nabla_j \mathbf e _i = \Gamma^k_{\ \ ij}\mathbf e_k$

I saw in other places the Christoffel symbols defined so

$\partial_j \mathbf e _i = \Gamma^k_{\ \ ij}\mathbf e_k$

2) Is the covariant derivative of basis vectors the same as the regular derivative of a basis vector?or are these just two different definition of the Christoffel symbols?

Another contradiction I saw is that they write the following formula:

in the end of the section "Coordinate Description"

where you add here a Gamma for each upper index and subtract a Gamma for each lower index according to the rule written there.

According to this it seems to me that:

$\nabla_j \mathbf e _i = \partial_j \mathbf e _i - \Gamma^k_{\ \ ij}\mathbf e_k$

Which is also inconsistent with how they defined the covariant derivatove

3) Is this a contradiction or a confusion of mine?

Thank you very much, sorry it's so long

If it's a problem I can break the question up into two questions or something

Answer

1) The confusion comes from an omission of parentheses in these notations. In the first case we do indeed have $$\nabla_{\vec{e}_i}(u^j) = \frac{\partial u^j}{\partial x^i},$$ since $u^j$ is just one non-specific component of $\vec{u}$. In the second case, they mean to take the component after differentiating the tensor: $$u^j{}_{;i} = (\nabla_{\vec{e}_i} \vec{u})^j = \frac{\partial u^j}{\partial x^i} + u^k \Gamma^j_{ki}.$$ I am using arrows instead of Roman type to indicate vectors in order to emphasize which things are full vectors (which may have subscripts, e.g. $\vec{e}_i$ is the $i$-th vector in your basis) and which things are components.

2) There should only be one set of Christoffel symbols. In what context was this the definition?

Also, covariant derivatives reduce to partial derivatives on scalars.

3) The confusion here comes from the use of $i$ in $\vec{e}_i$ as a label on which basis vector is being used, rather than on which component of a given vector is in place. Think of $\vec{e}_i$ as one symbol, such as $\hat{x}$ or $\hat{y}$. (This is indicated by the Roman as opposed to Italic typeface in the question, which again I've switched to an arrow to draw attention to the vector nature of the symbol.) We use lower subscripts so they don't interfere with the upper superscripts that would label the components. That is, $\vec{e}_i$ has components $e_i^0$, $e_i^1$, etc. As an object whose components are indexed with upper indices, one uses a positive Christoffel term: $$(\nabla_j \vec{e}_i)^k = \partial_j e_i^k + \Gamma^k_{jl} e_i^l.$$ Note that $e_i^k = \delta_i^k$, which is a constant and therefore has vanishing partial derivative. Contracting the Christoffel symbol with the Kronecker delta in the second term leaves only $\Gamma^k_{ji}$, as expected.

special relativity - Is causality a formalised concept in physics?

I have never seen a “causality operator” in physics. When people invoke the informal concept of causality aren’t they really talking about consistency (perhaps in a temporal context)?

For example, if you allow material object velocities > c in SR you will be able to prove that at a definite space-time location the physical state of an object is undefined (for example, a light might be shown to be both on and off). This merely shows that SR is formally inconsistent if the v <= c boundary condition is violated, doesn’t it; despite there being a narrative saying FTL travel violates causality?

Note: this is a spinoff from the question: The transactional interpretation of quantum mechanics.

experimental physics - Gravitational waves detection, any news?

Is the detection of gravitational waves a reality with nowadays technology? Are there recent news?

Answer

Unfortunately gravitational waves have not been detected yet.

There is a number of Earth-bound detectors planned and already in operation (e.g. LIGO, Geo 600, Virgo, Nanograv and others). As for space-borne detectors, ESA works on Next Gravitational-Wave Observatory after NASA pulled out of LISA project in April 2011 due to funding problems. Joint NASA/ESA mission, LISA Pathfinder will launch in June 2013 testing technologies to be used by NGO.

Keep an eye on the pages and blogs of these projects if you'd like to stay up to date on their progress. Also, if gravitational waves are detected, the discovery will no doubt be announced here and even here.