All Questions
200
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End points of event horizon
I am reading The Nature of Space and Time by S. W. Hawking. In the last paragraph on page 16 he said that:
event horizon may have past end points but don't have any future end points
I understand ...
- 59
2
votes
0
answers
36
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How does loop quantum gravity handle spacetimes which aren't globally hyperbolic, like the Kerr metric?
Loop quantum gravity assumes spacetime is globally hyperbolic. However, the interior of a Kerr black hole isn't globally hyperbolic, containing closed timelike curves. So, how are Kerr black holes ...
- 31
2
votes
1
answer
79
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A few doubts regarding the geometry and representations of spacetime diagrams [closed]
I had a couple questions regarding the geometry of space-time diagrams, and I believe that this specific example in Hartle's book will help me understand.
However, I am unable to wrap my head around ...
- 23
2
votes
1
answer
89
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Confusion about timelike spatial coordinates
I'm pretty new to general relativity, and I'm self-studying it using Sean M. Carroll's text on the subject. In Section 2.7, he introduces the notion of closed timelike curves. He gives the example of ...
- 160
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votes
1
answer
79
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How to Understand Negative Energy in the Ergoregion?
I am trying to understand the Penrose process and having trouble explaining negative energy in the ergoregion.
How I interpret it is:
Energy is the dot product between the four momentum of the object ...
- 63
2
votes
1
answer
44
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Are non-point spacetime events partially ordered?
When describing events in spacetime, we usually use points. We then phrase the relation between points as a trichotomy: either they are timelike, spacelike, or lightlike separated, based on the ...
- 180
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0
answers
91
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Is the causal structure completely determined by the Weyl tensor alone?
By causal/conformal structure I mean the context of Malament's 1977 theorem. If I understand correctly this means that any two spacetimes which agree about all of the future-directed continuous ...
- 151
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vote
2
answers
71
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How do I interpret the time axis in a diagram with multiple light cones?
Light cones are often drawn on a spacetime diagram that has a directional time axis like the fourth one on this page:
There is a time axis, and all of the light cones are align with it because this ...
- 113
3
votes
4
answers
364
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Regarding the signature of special relativity
in special relativity we add time as a dimension and replace euclidean space $ \mathbb{R}^4 $ with a pseudo-euclidean space $ \mathbb{R}^{1,3} $ of signature $ (1,3) $ by defining a quadratic form $\...
- 309
1
vote
1
answer
164
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How does null infinity differ from ordinary infinity?
Null infinity is the diagonal lines on the edge of a Penrose diagram. It seems to be the place where beams of light go if they never bump into anything, but only light can go there. It appears to be ...
- 278
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2
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65
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Unidirectionality of Time in Spacetime
I have a question regarding the dimension of time. We all know that an event in spacetime is defined by a point
$$ {x}^{u} = (ct, x, y, z) .$$
The only component that breaks the symmetry is $ct$, ...
- 151
-6
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2
answers
131
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Do the Lorentz transformations contradict reality by implying time had no beginning? [closed]
Consider the following two points, or events as they are more commonly called, in SpaceTime:
Event 1: $(x,t) = (0,0)$
Event 2: $(x,t) = (a,0)$
Take t=0 to correspond to the first moment in time.
As ...
- 179
2
votes
0
answers
35
views
Is there a general methodology for causal nets of observables regardless of kinematics?
The typical definition of a causal net of observables in quantum theory is to consider, for the case of a (globally hyperbolic) spacetime $M$, the category of open sets $O(M)$ ordered by inclusion, in ...
- 16.5k
12
votes
5
answers
2k
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Could relativity be consistent if there are multiple light-like fields with different invariant speeds?
My understanding of real physical theory of electromagnetism goes like this:
The Maxwell equations can be used to derive the speed of light;
$$\nabla\cdot\textbf{E}=0$$
$$\nabla\cdot\textbf{B}=0$$
$$\...
- 5,148
1
vote
1
answer
109
views
Carter-Robinson Theorem
There are uniqueness theorems that classify Black holes according to its mass, angular momentum and charge. One of the theorem is Carter-Robinson theorem which has many assumptions and then it says ...
- 59
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1
answer
54
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In Relativity theory, is chronological relation an order relation?
Let $(M,g)$ be a (Lorentz) spacetime, i.e a connected smooth manifold $M$ with a metric tensor field $g$ and a time orientation called future direction which is defined by a smooth timelike vector ...
- 143
1
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0
answers
44
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Global Hyperbolicity and Timelike Boundary
I am trying to understand and show that asymptotically Anti-de Sitter spacetimes are not globally hyperbolic.
Now, I have found papers that talk about global hyperbolic spacetimes with timelike ...
- 743
3
votes
1
answer
45
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Counterexample to the observable algebra of a region and its causal completion being the same
I was reading a paper by Ed Witten called "Algebras, Regions and Observers". It can be found here: https://arxiv.org/abs/2303.02837
A major theme is theorems relating the algebra of ...
2
votes
0
answers
63
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Conjugate points on manifolds
My question is:
Why do conjugate points exist on globally hyperbolic manifolds, satisfying the strong energy condition?
We define M to be globally hyperbolic if it posseses a cauchy surface and a pair ...
user390166
1
vote
2
answers
132
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Do events very far away happen in a different timeline?
I am not sure how to ask this question in a concise manner so I am sure somebody out there explained it but I cannot seem to find it.
So I recently watched some videos explaining that $c$ not only ...
- 119
2
votes
2
answers
161
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If I were to drop my phone into a black hole, would I be able to catch it?
Say, for the sake of argument, I am outside the event horizon of a black hole and accidentally drop my phone (or some other object) into the hole. If I were to enter the black hole, would I ever be ...
- 121
3
votes
2
answers
612
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Null infinity reachable by timelike worldlines?
Usually, Penrose diagrams are marked with points and segments being named past/future timelike infinity $i^{-,+}$, past/future null infinity $\mathscr{I}^{-,+}$ and spacelike infinity $i^0$ -- see for ...
- 743
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28
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Same curvature but different orientation of light cones? [duplicate]
Can there be two regions of spacetime which have the same curvature, but with their light cones oriented in different directions?
In the Stack Exchange question "General Relativity via light ...
1
vote
1
answer
90
views
What does hypersurface of simultaneity exactly mean?
HSS - "Hyper Surface of Simultaneity"
Listening to different sources online I understood that HSS for a observer represents the points that are at same moment of time.
Consider a 1d world. ...
7
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2
answers
2k
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Do the Einstein Field Equations force the metric to be Lorentzian?
In GR, we are working with Lorentzian metrics, which are examples of a pseudo-Riemannian metrics. That is, we are trying to find pseudo-Riemannian $g_{\mu\nu}$ that are solutions to the field equation ...
- 453
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votes
1
answer
85
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Find an example of a closed, achronal set $S$ in Minkowski spacetime such that $J^+(S)$ is not closed
This is one of the exercises on Wald's General Relativity:
Chapter 8, Problem 8.b Find an example of a closed, achronal set $S$ in Minkowski spacetime such that $J^+(S)$ is not closed. (Hint: ...
- 22.1k
2
votes
2
answers
156
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Relationship between spacelike and timelike distances in General Relativity vs. Special Relativity
In Minkowski spacetime, the distance $d_S$ between two space-like separated events $x$ and $y$ can (up to constant) be given by a distance between the two time-like separated events $z$ and $w$ where $...
- 125
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0
answers
93
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Spatial separation in analogy to time separation in Lorentzian geometry?
O'Neill (Semi-Riemannian Geometry With Applications to Relativity, 1983, p. 409) defines time separation between two events as follows:
"If $p, q \in M$, the time separation $\tau(p, q)$ from $p$...
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-1
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1
answer
55
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Since we observe stars, galaxies, etc in their past - is it correct to say that our present is only present for us but is in the past from afar? [closed]
So - I know that when we observe galaxies, we are observing their past. And that if the same past-them were to be looking at us, they would see our past. But when we look at their past - that is a ...
-1
votes
1
answer
129
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According the theory of general relativity, what is the role of causality in the changes of the curvature of spacetime? [closed]
In Einstein's equations the curvature of spacetime and energy-momentum-pressure density are correlated. Is it clear when changes in matter energy density affect causally to curvature and when changes ...
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90
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Geometrically Impossible Spacetime
A result in math says that $S^n$ carries a Lorentzian metric iff $n$ is odd.
Using it we can observe that a 2-sphere spacetime is impossible, a 3-sphere spacetime is geometrically possible, but again ...
1
vote
1
answer
82
views
What is a "timelike half-curve"?
I know what a timelike curve is. But what is a time-like half-curve, as in the definition of a Malament-Hogarth spacetime (below), which appears in this paper?
Definition: A spacetime $(M,g)$ is ...
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1
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1
answer
67
views
Is pp-wave spacetime strongly causal?
Is pp-wave spacetime, strongly casual?
If not, where is it on the causal ladder?
- 1,268
2
votes
1
answer
185
views
No inconsistencies due to closed timelike curves in General Relativity
Is there anything wrong with the following argument ruling out inconsistencies with closed timelike curves?
Math
A solution in the context of General Relativity is a pseudo-Riemannian manifold $(M, g)$...
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1
answer
44
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A question on the causal hierarchy/ladder and the existence of CTC
What does the existence of CTC imply for the Causal Structure of the spacetime?
Can a strongly causal spacetime have any CTC[without fluctuating the metric]?
Is there any such example?
Can someone ...
- 1,268
5
votes
1
answer
209
views
Why should a Cauchy surface be closed?
A Cauchy surface is defined on any spacetime $M$ as a subset $S$ which is closed, achronal, and whose domain of dependence $D(S) = M$.
Why do we include the "closed" condition in the above ...
- 1,588
3
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0
answers
129
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I need help with a proof in Hawking & Ellis [closed]
Here's a proof in Hawking and Ellis (1973) of proposition 6.4.6:
The definition of "strong causality" used in the book is that for every point $p$ and every neighborhood $U$ of $p$, there ...
- 863
1
vote
1
answer
63
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Non-Compactness in Penrose Singularity
I've been studying singularities in GR, and (obviously), came across PST.
Let us state it as the following:
Let $(M, g)$ be a connected globally hyperbolic
spacetime with a noncompact Cauchy ...
- 487
1
vote
1
answer
111
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Topology of Time
I came across the concept of topology of time and causality in Reichenbach book, "Philosophy of Space and Time". It would be nice to have list of references of recent developments of the ...
0
votes
2
answers
54
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Time-Like Separated Events: Can there be more than one reference frames for which 2 events can happen at same place?
For time-like separated events, reading of 2nd event of any observer from central event (i.e. at origin)
should lie within top light cone (we will consider only future events and ignore the bottom ...
- 1,481
1
vote
1
answer
96
views
When a curve is future (past) inextendible?
Future (past) endpoint: We say that $p\in M$ is a future (past) endpoint of a curve $\lambda$ if for every neighborhood $O$ of $p$ there exists a $t_0$ such that
$\lambda(t)\in O$ for all $t>t_0$ (...
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3
votes
1
answer
530
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Confusion regarding Geodesics
Suppose we have a causal curve and we can cover the causal curve by convex normal neighborhoods. We also know that, in convex normal neighborhood there will exist a unique geodesic inside the ...
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1
answer
114
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Why is causal order not related to directionality of time?
Hans Reichenbach argues for the causality and causal chain to define a topological coordinative definition of time order. Here is an excerpt from his textbook, The Philosophy of Space and Time, Dover(...
- 625
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1
answer
112
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How to show mathematical equivalence between the idea of relativistic mass and the geometric explanation of why massive objects can't reach $c$?
I've frequently seen two different explanations for why, in SR, it's impossible for an massive object to reach $c$:
As a massive object approaches $c$, its kinetic energy starts being converted to ...
- 2,327
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votes
0
answers
84
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Conformal Diagram for Astrophysical Black Hole
I have a question about the conformal diagram of an ‘astrophysical’ black hole which forms in finite time (but with no evaporation).
Usually I see the conformal diagram presented as something similar ...
- 55
0
votes
1
answer
69
views
Minkowski spacetime
I have recently become interested in special relativity. I would like to stress that I am not a physicist, but just a curious person. I read that a piecewise twice continuously differentiable curve $\...
- 186
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votes
2
answers
93
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A question on negative time and Minkowski Space
Can someone please explain why a point originating inside the cone can never be the cause of an event outside the cone? Is there a real life example of this? The only thing I can think of is from a ...
- 2,706
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1
answer
101
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Doubt about a simple equation in Srednicki's QFT textbook
On page 30, he writes ($ Px = P^\mu x_\mu = \textbf{P} \cdot \textbf{x} - H t $)
$$ e^{-i P x /\hbar } \varphi(0) e^{i P x /\hbar } = \varphi(x) ,\tag{2.21}$$
which he states is a relativistic ...
- 1,957
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votes
0
answers
55
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Why is the edge of a closed achronal set equal to the edge of its future Cauchy horizon?
This question is related to Proposition 6.5.2 of Hawking & Ellis. It states that
$$\text{edge}(H^+(\mathscr S))=\text{edge}(\mathscr S),$$
for a closed achronal set $\mathscr S$.
Of course, ...
- 1,487
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votes
2
answers
107
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Light-like interval and simultaneity [closed]
Why if the interval between two events is light-like then there is no frame of reference where the events occur at either the same time? If I assume that the 2 events happen at the same time, I arrive ...
