# Questions tagged [spacetime-dimensions]

Use this tag for dimensions of a manifold, typically the space-time. DO NOT USE THIS TAG for dimension of a physical quantity nor for the size of an object.

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### Question on spatiotemporal dimensionality about the contradictions of time being a dimension

We can axiomatically see that all spatial dimensions have a fundamental rule where they can either move back or forwards infinitely. However, the temporal dimension started when the universe began and ...
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### What happens if we differentiate spacetime with respect to time? [closed]

Essentially, what would differentiating space-time with respect to time provide us with? What are the constraints associated with such operations? Is it possible to obtain a useful physical quantity ...
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### Do the Komar/ADM mass equations also hold in 2+1D?

All definitions I have come across for the ADM mass require asymptotic flatness, which always is defined for 4 dimensional spacetimes. I was wondering if these formulae in 3+1D hold in 2+1D aswell?
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### Multiple time dimensions in the eternal inflation model

From a lecture by Prof. Kaiser, I reckoned that according to the Eternal Inflation model, it is possible that all of the 10500 topologies posited by string theory could exist somewhere in the region ...
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### Definition of the gravitational constant in 1+1 gravity

In this paper, the author formulates a $(1+1)$-dimensional theory of gravity by taking the trace of the Einstein equations $$\left(1 - \frac{D}{2}\right)R = 8\pi G_D T,\tag{2}$$ (where $G_D$ is the ...
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### Poincaré algebra and supersymmetric spaces

If i understand correctly, a supersymmetry algebra should contain as a subalgebra the Poincaré algebra, however for a supersymmetry algebra the corresponding supersymmetric (Minkowski) space has ...
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### What does the superscript $3$ in $d^3x$ mean in an integral?

At the risk of seeming ignorant, please explain what does the superscript $3$ in $d^3x$ mean in the integrals 5.12, 5.13, 5.14, 5.15? Why 3? Why there is no such in the 5.10 integral?
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### How does 1D Schrödinger equation arise out of the postulated 3D Schrödinger equation and solving 1D particle using 3D Schrödinger equation?

I've stumbled upon this question when I was trying to solve the Schrödinger equation for a particle confined to a 1D line with some given time independent potential $V(x)$. The energy eigenstates ...
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### How many dimensions are in string theory? [duplicate]

How many dimensions are in string theroy? I heard that there are 11 but to my understanding, there is an infinite, also can strings be on a 2D plane?
1 vote
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### Does the distance between two objects of mass not matter when measuring strength of gravity in one-dimensional space?

From all that I have heard about Newton's Law of Universal Gravity, one fact, which I find quite interesting, is that the distance between the two objects of mass is squared and not cubed due to our ...
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### What are dimensions and how are they defined? [duplicate]

We all study dimensions as a topic in physics in which we are taught the dimensions of different physical quantities but I don't understand what is the connection between the things that we study ...
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### Is there a true one-dimensional object? [closed]

I'm reviewing and expanding my knowledge of dimensions. We live in three spatial dimensions but, apart from volume, we also have the concept of surface and curve. However, if you write a line on paper,...
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1 vote
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### How to prove that the Brachistochrone problem could be reduced to finding a curve on a plane?

Given two points in space, the 2D Brachistochrone problem could be solved to give solution of a cycloid. I am wondering how could one prove that in arbitrary dimensions ($d\geq 3$) with a 1D uniform ...
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### Can you naïvely reduce the dimensionality of a QFT?

I want to study a QFT, given by an action $S$ which is defined in $(2+1)$ dimensions, i.e. $$S=\int d^3x \mathcal{L}[\phi,\,\partial\phi].$$ This QFT is invariant under rotations, i.e. in radial ...
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### Relation between the number of curvature functions and dimensions in GR

I am reading Weinberg's Gravitation and Cosmology. On page 10, it reads In $D$ dimensions there will be $D(D+1)/2$ independent metric functions $g_{ij}$, and our freedom to choose the $D$ coordinates ...
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### Electromagnetism in 2+1 dimensions?

Consider the Lorentz group $SO(2,1)$ in 2+1 spacetime dimensions. It's little group for massless particles should be just "$SO(1)$", which is just a trivial group with an identity element. ...
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### Mechanics/Statics: How to decide which statics problem can be modeled/solved in 2D or 3D? What are the steps to identify the dimension?

I am a first year mechanical engineering student. In statics we learn to solve/model different problems (free body diagram, sum of forces in $x/y$ etc...) in 2D and in 3D. But how to think about 2D? ...
1 vote
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### Action formalism of braneworld gravity and effective field equation on the brane

Is it possible to derive the effective gravitational field equation on the brane by simply varying the action? Context: The popular way to derive that equation is by starting from Einstein's field ...
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### Tensor densities in 1 dimensional space

When we consider a 1 dimensional manifold, is a scalar density with weight (-1) the same as a covector? In particular, in a theory of gravity, if we consider $\sqrt{-g}$, with $g=\det(g_{\mu \nu})$, ...
1 vote
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### The Lebesgue covering dimension of the Cosmic String interval topology

Take the spacetime $(M,g)$ that satisfies Einstein's Field Equations exactly where $g$ is locally: $$g= - c^2 dt^2 + d \rho^2 + (\kappa^2 \rho^2 - a^2) d \phi^2 - 2 ac d\phi dt + dz^2 \$$ in the ...
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### How small can we measure space? [closed]

I got this question after looking into transcendental numbers and I noticed how there are some distinctions that should be made from numbers and reality especially in measurement of length for example ...
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### 3D manifestation of a higher dimensional object

The starting points of this theorical exploration are the following. I do believe we exist in a universe where 10 (or 11) dimensions do exist, but the ones beyond 3 spatial + 1 time are compactified. ...
1 vote
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### Scalar spherical harmonics in $S_n$

In the Kaluza klein reduction we can "decompose" the spacetime $M_n$ as $M_n = M_4 \otimes K_d$, in which $K_d$ is a compact spacetime. So, functions like a scalar $\phi(x,y)$ can be ...
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### Rotational states in higher dimensions: multiple magnetic quantum numbers

In 4 dimensions, arbitrary rotations are usually double rotations (rotations which can be understood as happening independently on two different planes with different rotation angles). It certainly ...
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### Magnets in 2 Spatial Dimensions?

In 2+1 dimensions of spacetime, the electromagnetic field is made up of a vector electric field and a scalar magnetic field. At each point in space, there is a magnetic field value, which we can ...
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### Centre of mass of particles in different dimensions

Can we find the centre of mass of a particle in 4 dimensions? Can we find it in more than four dimensions?
1 vote
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### What's the Newtonian potential in 2+1 gravity?

I understand that there are no propagating degrees of freedom (i.e. gravitational waves) in 2+1 dimensions. There are a couple of arguments to show this. One is to count degrees of freedom of general ...
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### Why consider more than 3 dimensions in the Ising model? [duplicate]

Are there real-world physical systems to which higher dimensional ($d>3$) Ising models correspond?
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### Extension of a pattern from electrostatics into more than 3 dimensions [duplicate]

I am taking an introductory physics course, and the chapter we are on is about electrostatics. One section of our textbook has talked about the electric field generated by a charged object that is ...
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### Do we live in a 4-dimensional space, i.e. a 5-dimensional spacetime? [duplicate]

As far as we know: If two one-dimensional lines are placed parallel, they need to be on a two-dimensional plane. If two 2-dimensional planes are placed parallel, they need to be in a 3-dimensional ...
1 vote
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### What is special about conformal field theory in 2d? [duplicate]

In the most of textbooks about CFT, the special case of 2d is noticed in which complex coordinates play important role and it reads some results like the conformal transformation of energy-momentum ...
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### How many independent equations do Maxwell's equations represent in arbitrary dimensions?

In an arbitrary number of spacetime dimensions $D$, Maxwell's equations are \begin{align*} \mathrm{d}F &= 0, \\ \mathrm{d}(\star F) &= -J. \end{align*} How many independent equations does this ...
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1 vote
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### Is there any explicit theoretical application of Brouwer's Topological Invariance of Dimension theorem?

I'm interested in applications of Brouwer's Topological Invariance of Dimension theorem. I study mathematics but know very little about physics, but I imagined that the Invariance of Dimension theorem ...
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### Higher dimension observer

In Quantum Mechanics (double slit electron experiment) a third dimension observer could only see two kinds of patterns in a screen: -Two lines behind each slit if one chooses know the electron ...
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### What exactly are the Differential Forms in Maxwell's Equations? [duplicate]

While trying to understand Maxwell's equations (having learned a bit about manifolds) I encounter the following issue: Gauss' Law seems to integrate the electric field $E$ over a $2$-manifold, ...
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### What's the definition of spin for a particle in $d$-dimensional Minkowski spacetime?

Consider a relativistic quantum theory in d-dimensional flat spacetime. Neglecting possible internal symmetries, a particle is defined as a system whose Hilbert space furnishes the support of an ...
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### Is there an "escape velocity" for closed dimensions? [closed]

Assuming a closed universe, the shape of the universe is often considered, or at least presented in pop science, to be a glome (4-sphere), and popularly depicted as behaving analogously to a 3-sphere, ...
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### What is the minimal count of dimensions in which our curved 3D space can be embedded into? [duplicate]

As I know, GR does not need to assume anything about a >3D space, where our 3+1 spacetime can be embedded into. However, I think the curved 3D space (more clearly, the spacelike cuts of the 3+1D ...
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### Why does a degree of freedom vanish from 3D to 2D in that tensor construction?

Let's assume an arbitrary tensor in 3D coordinates: $g_{ij}$ with $i, j$ in $[1,3]$. It shall be arbitrary, meaning not symmetric. It has 9 entries which equals 9 degrees of freedom (dof). Now, I ...
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### What is the angle of gravity? [duplicate]

Let's have two objects touching each other, i.e. me standing on the earth. We propel the smaller object directly away from the larger, i.e. I jump. The objects move apart, slow down and then return ...
1 vote
One way I know to get intuition for the derivation of the force equation $$F=\frac{GM_1M_2}{r^2}$$ in Newtonian Mechanics is to imagine gravitational field lines, in combination with certain ...