Tipler 1976: Clarifying Symbol Meaning

[hawkdron496]

TL;DR

A request for notational clarification in an old paper

I'm reading Tipler's 1976 paper, "Causality Violation in Asymptotically Flat Spacetimes" and he keeps using a symbol which seems to resemble the symbol for Future Null Infinity in a strange font, but it's usage doesn't make sense with what I would expect if that's what the symbol meant. He doesn't define it anywhere in the paper, so I assume it must be standard notation that I'm missing. The symbol in question is the one that in the first photo we're interested in the causal past of, and in the second photo, we're evaluating at the point q.

 

[Ibix]

$J^-(X)$ is the region visible from $X$, so $J^-(\mathfrak{I}^+)$ would be the region visible from infinity, which matches up with the text you quote. What's the problem?

 

[hawkdron496]

$J^-(X)$ is the region visible from $X$, so $J^-(\mathfrak{I}^+)$ would be the region visible from infinity, which matches up with the text you quote. What's the problem?

I'm not clear on what it means when he takes $\mathfrak{I}^+(q)$. Is it just the set of null rays going off to infinity that pass through q?

 

[Ibix]

I'm not clear on what it means when he takes $\mathfrak{I}^+(q)$. Is it just the set of null rays going off to infinity that pass through q?

Could be, or maybe the portion of future null infinity that's in the future of $q$. Not sure. I'll look in Wald later if nobody else answers you first.

Do you have a link to Tipler's paper?

 

[vanhees71]

https://doi.org/10.1103/PhysRevLett.37.879

 

[Ibix]

Paywalled, unfortunately, but thanks.

 

[Ibix]

Ok, can't find any use of $\mathfrak{I}^+(q)$ in Wald. It might just be a typo for $I^+(q)$, which is the interior of the future lightcone of event $q$? Difficult to comment without more context.

 

[robphy]

If I recall, it says that it follows the notation of Hawking and Ellis.

For those without access to the article, it might be good (as @Ibix suggests) to give more context of the questionable notation…. That is, give the whole sentence or paragraph where it is used. Thus, one can match up the definitions, then the notations.

 

[hawkdron496]

Will do: the relevant paragraph is here:

The questionable notation is in the Proof section of the proposition.

 

[Ibix]

I presume $\tilde{D}^+$ is the closure of $D^+$? I'm also not sure what $\phi$ is supposed to be, unless it's meant to be the empty set symbol $\emptyset$. If so, I think $\mathfrak{I}^+(q)\cap\tilde{D}^(S)=\phi$ makes sense in context if interpreted as ${I}^+(q)\cap\tilde{D}^(S)=\emptyset$ - i.e. if $\mathfrak{I}^+$ is a typo for $I^+$.

If anyone disagrees with me they're probably right - I'm literally doing this with my phone in one hand and Wald in the other.

 

[hawkdron496]

I presume $\tilde{D}^+$ is the closure of $D^+$? I'm also not sure what $\phi$ is supposed to be, unless it's meant to be the empty set symbol $\emptyset$. If so, I think $\mathfrak{I}^+(q)\cap\tilde{D}^(S)=\phi$ makes sense in context if interpreted as ${I}^+(q)\cap\tilde{D}^(S)=\emptyset$ - i.e. if $\mathfrak{I}^+$ is a typo for $I^+$.

If anyone disagrees with me they're probably right - I'm literally doing this with my phone in one hand and Wald in the other.

From what I've been able to find online, the symbol $\tilde{D}(S)$ is the same as the usual symbol for domain of dependence, but only for timelike curves rather than any causal curve, according to this stack overflow post:

https://physics.stackexchange.com/q...pment-or-future-domain-of-dependence-why-is-d

The symbol is from Hawking and Ellis, apparently.

 

[hawkdron496]

So, that strange notation occurs again, later in the paper:

which makes it feel less likely that it's a typo.

 

[Ibix]

I agree it seems less likely to be a typo if it recurs. I'd get hold of a copy of Hawking and Ellis, then, if I were you.

 

[hawkdron496]

I agree it seems less likely to be a typo if it recurs. I'd get hold of a copy of Hawking and Ellis, then, if I were you.

Yep, I suppose I will. Thank you for the help.