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Questions tagged [metric-space]

A metric space characterizes a set of elements by means of one number value for each pair of elements. (Additional conditions may apply.) It formalizes and broadens the idea of geometric distances, as for instance between particular cities.

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Expressing curvature invariants ($K_1, I_1, ... $), at any one event, through Synge's WF $\sigma$ (given of each event pair, in a suitable region)

Considering a set $\mathcal S$ of events such that for each pair $p, q \in \mathcal S$ Synge's world function $\sigma$ is defined and the corresponding value $\sigma[ ~ p, q ~ ]$ is given, and such ...
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Is spacetime isomorphic to a metric space?

I know that spacetime, as described by General Relativity (GR), is a pseudo-Riemannian manifold. The label "pseudo" is due to the fact that the metric of spacetime entails not only positive ...
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Is there a standard name for Robb's spacetime invariant equation?

I'm not sure exactly how to categorize Robb's treatment of the spacetime interval. But it seems like a gem of simplicity and insight. The following illustrations are based on MTW Box 1.3. As drawn, ...
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What exactly is A. A. Robb's version of "the result first proved by Robb (1936)"?

Considering four distinct events in a spacetime region in which values of Synge's world function $\sigma$ are defined (up to a common non-zero factor) for each pair of events, and specificly ...
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Open sets in Minkowski spacetime

I don't know how to imagine open sets in Minkowski spacetime. I have seen that there are many diffrent ways of constructing them — that's OK. But for example. which construction do people mean in the ...
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Can the image of a spacetime geodesic be characterized through Synge's world function?

Since a geodesic is understood to be a map $\gamma : \text{real number interval} \rightarrow \text{spacetime } \mathcal S$, with certain additional properties, the image of a particular geodesic is ...
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Why is the time coordinate different in the metric? [duplicate]

I have been using the metric for quite a while now and I never thought about it. Why does the time coordinate always have an opposite sign to the space one? In other words, why does the metric have ...
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Clarification on statement in “Unitary Symmetry and Elementary Particles” by Lichtenberg

He says that: The set of values of the parameter or parameters which characterize a group element can be considered to be points in some kind of space. The number of parameters characterizes the ...
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How to express the magnitude of proper acceleration through spacetime intervals

Given the trajectory of participant $P$ in a flat region $\mathcal S$ of spacetime through the set of events $\mathcal E_P \subset \mathcal S$ in which $P$ had taken part, and given the values of ...
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How to identify an Euclidean space?

How do I determine that a certain space is Euclidean space or not? Are Spherical and Cylindrical Coordinates Euclidean too? This question may be be elementary but I need to understand this.
Samapan Bhadury's user avatar
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Why do we use dual space in some circumstances and inner product in others?

In undergraduate linear algebra, the concept of a dot product, generalized to the inner product on an inner product space, is introduced fairly early as a way to multiply 2 vectors together to get a ...
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Is the space on which the BRST $Q$ operator acts a Hilbert space?

When I looked at the BRST symmetry in Yang-Mills-Theories I was puzzled by the statement: Suppose we go back to canonical quantisation with a Hilbert space $\mathcal{H}$. The BRST symmetry leads to ...
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Given a Spacetime in terms of Lorentzian distance values, how to determine which pairs of events were spacelike separated?

The geometric relations between pairs of events of a spacetime $\mathcal S$ can generally be characterized by values of Lorentzian distance (cmp. J.K.Beem, P. Ehrlich, K.Easley, "Global Lorentzian ...
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Why is $g(v,v)$ the speed in general relativity?

if $v_{\gamma,p}$ is the velocity along a curve at point $p$ on the manifold of space time, and $g$ is the metric tensor, then $g(v_{\gamma,p},v_{\gamma,p})^{1/2}$, calculated in tensor notation by $(...
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How to express in terms of spacetime intervals the average speed of one participant having moved between two suitable others

Given a flat region of spacetime as set $\mathcal S$ of events together with values of spacetime intervals (up to a common non-zero constant) for each pair of events, $s^2 : \mathcal S \times \mathcal ...
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How to express in terms of spacetime intervals whether two participants in a flat region were at rest to each other

Given a flat region of spacetime as set $\mathcal S$ of events together with values of spacetime intervals (up to a common non-zero constant) for each pair of events, $s^2 : \mathcal S \times \mathcal ...
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Relativity and "light years"

As I understand relativity, time is relative to your velocity, meaning your watch moves slower relative to those who are stationary when moving at great speeds. So if that's true, then when we talk ...
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Can you recover the values of spacetime intervals $s^2$ from given causal relations between events?

Given a suitable set $\mathcal S$ of events together with their (pairwise) causal relations, i.e. for each pair of distinct events $\mathsf A, \mathsf B \in \mathcal S$ the assignment whether $\...
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What is the most general definition of a coordinate system?

What is the most general definition of a coordinate system? Specificly: given a suitably general metric space $(\mathcal S, s)$ consisting of a set $\mathcal S$ of elements (for instance: a set ...
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Questions about "geodesic path" and "arc length of a geodesic path" in the context of GTR and "Lorentzian manifolds"

I'd like to check my understanding of the notions "geodesic path" and "arc length of a geodesic path" in the context of GTR and "Lorentzian manifolds". Considering a set of "spacetime events", $\...
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How is the curvature of space measured on earth scale?

How do we measure the curvature of space due to earths gravity since we can't dig a hole through the centre of the earth and measure the actual diameter of earth which is slightly different (1.8cm) ...
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Do three spacelike separated events on the same lightcone satisfy the triangle inequality?

If three events are (pairwise) spacelike related to each other then their interval ratios do not necessarily satisfy the triangle inequality. (For example, consider the case that two of the three ...
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Can isotropy (or anisotropy) be expressed in terms of intervals ($s^2$) between pairs of events?

Considering a set $\mathcal S$ of events and given the values of intervals $s^2[~P, Q~] \in \mathbb R$ for all pairs of events $P, Q \in \mathcal S$ (up to a common non-zero scale factor): how can ...
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Given two events such that either one of them is 'on the light cone' of the other, do they constitute a 'null interval'?

One basic part of determining a metric (or applicable generalization) of a given set $\cal S$ of events (up to an arbitrary non-zero constant) is to determine to which pairs among those events, $\...
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