# What is Minkowski space: Definition and 50 Discussions

In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity.Minkowski space is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime between events. Because it treats time differently than it treats the 3 spatial dimensions, Minkowski space differs from four-dimensional Euclidean space.
In 3-dimensional Euclidean space (e.g., simply space in Galilean relativity), the isometry group (the maps preserving the regular Euclidean distance) is the Euclidean group. It is generated by rotations, reflections and translations. When time is amended as a fourth dimension, the further transformations of translations in time and Galilean boosts are added, and the group of all these transformations is called the Galilean group. All Galilean transformations preserve the 3-dimensional Euclidean distance. This distance is purely spatial. Time differences are separately preserved as well. This changes in the spacetime of special relativity, where space and time are interwoven.
Spacetime is equipped with an indefinite non-degenerate bilinear form, variously called the Minkowski metric, the Minkowski norm squared or Minkowski inner product depending on the context. The Minkowski inner product is defined so as to yield the spacetime interval between two events when given their coordinate difference vector as argument. Equipped with this inner product, the mathematical model of spacetime is called Minkowski space. The analogue of the Galilean group for Minkowski space, preserving the spacetime interval (as opposed to the spatial Euclidean distance) is the Poincaré group.
As manifolds, Galilean spacetime and Minkowski spacetime are the same. They differ in what further structures are defined on them. The former has the Euclidean distance function and time interval (separately) together with inertial frames whose coordinates are related by Galilean transformations, while the latter has the Minkowski metric together with inertial frames whose coordinates are related by Poincaré transformations.

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1. ### I Approximate local flatness = Approximate local symmetries?

Pseudo-Riemannian manifolds (such as spacetime) are locally Minkowskian and this is very important for relativity since even in a highly curved spacetime, one could locally approximate the spacetime into a flat minkowski one. However, this would be an approximation. Perhaps this is a naive...
2. ### I What is the relationship between honeycombs and Minkowski space?

Anyone know what honeycombs "tile" Minkowski space?
3. ### Time dilatation between straight and curved lines in Minkowsi space

Summary:: Special relativity - 2 astronauts syncronize their clocks and moves in different paths at different velocities, which clocks is left behind? and why? Hi everyone, i have the following problem and I'm not understanding if my strategy to solve it is correct: Two astronauts synchronize...
4. ### I Relativistic quantum mechanics

Given that the Minkowski metric implies the Lorentz transformations and special relativity, why do the equations of relativistic quantum mechanics, i.e., the Dirac and Klein-Gordon equations, require a mass term to unite quantum mechanics and special relativity? Shouldn't their formulation in...
5. ### I Understanding Pseudo-Angles in Minkowski Space: A Geometric Interpretation

Does the concept of the angle between two vectors make sense in Minkowski space? Does the concept of orthogonal basis for Minkowski space make sense? If it does, how is it defined? When we start with the usual (time, distance) basis for 2-D Minkowski space, the axes as drawn make a right...
6. ### I Hyperbolic Geom of Minkowski Space: Chung et al. 2009

In "The Geometry of Minkowski Space in Terms of Hyperbolic Angles" by Chung, L'yi, & Chung in the Journal of the Korean Physical Society, Vol. 55, No. 6, December 2009, pp. 2323-2327 , the authors define an angle ϑ between the respective inertial planes of two observers in Minkowski space with...
7. ### I QFT of Gravitons in Minkowski Space vs GR

A central feature of classical GR that it is background independent and operates via a curvature in space-time. As I understand it, this is not true of the other Standard Model forces which are consistent with special relativity and operate in Minkowski space, in which forces are transmitted via...
8. ### I I Came Up with a CTC in Minkowski Space: Exploring To What End?

If you viewed my most recent thread before this one, then you know that I have been studying curves in spacetime (timelike/spacelike/lightlike), and I have especially been looking into the CTCs (closed timelike curves) that the Godel metric is famous for. During my studies I found that I had to...
9. ### I Vectors in Minkowski Space & Parity: Checking the Effect

It is known that vectors change them sing under the influence of parity when ##(x,z,y)## change into ##(-x,-z,-y)## $$P: y_{i} \rightarrow -y_{i}$$ where ##i=1,2,3## But what about vectors in Minkowski space? Is it true that $$P: y_{\mu} \rightarrow -y_{\mu}$$ where ##\mu=0,1,2,3##. If yes how...
10. ### B Geodesic equation in Minkowski space clarification

Hi, So the geodesic equation is saying in my frame of reference I may see acceleration and then in your frame of reference you may see gravity? So by just changing coordinates you can create a "force" ? And also is this relevant to the Minkowski space or do I need to be in GR to be able to get...
11. ### I Line Element in Minkowski Space: Geometric Meaning

Hello, how can you imagine the geometrically meaning of the minus sign in ds2=-dx02+dx12+dx22+dx32, maybe similar to ds2=x12+dx22 is the length in a triangle with the Pythagoras theorem?
12. ### I Levi-Civita symbol in Minkowski Space

I set eyes on the next formulas: \begin{align} E_{\alpha \beta \gamma \delta} E_{\rho \sigma \mu \nu} &\equiv g_{\alpha \zeta} g_{\beta \eta} g_{\gamma \theta} g_{\delta \iota} \delta^{\zeta \eta \theta \iota}_{\rho \sigma \mu \nu} \\ E^{\alpha \beta \gamma \delta} E^{\rho \sigma...
13. ### A Delta function in continuation back to Minkowski space

The Green's function for a scalar field in Euclidean space is $$(2\pi)^4\delta^4(p+k) \frac{1}{p^2+m^2}$$ however when I continue to Minkowski space via GMin(pMin)=GE(-i(pMin)) there's seems to be a sign error: (2\pi)^4\delta^4(-i (p+k)) \frac{1}{-p^2+m^2}=(2\pi)^4\delta^4(p+k)...
14. ### I Deriving Einstein's Relativity Postulates from Minkowski Spacetime

Does the constancy of the speed of light for all observers naturally emerge from the Minkowski spacetime metric? Do Einstein's two postulates of relativity emerge from the Minkowski spacetime metric? Suppose we begin with Minkowski spacetime and the Minkoswki metric...
15. ### Stability of Minkowski space in semiclassical gravity?

Wald, General Relativity, p. 411, says that Minkowski space is unstable in semiclassical gravity. He gives a reference to this paper: Horowitz, "Semiclassical relativity: The weak-field limit," Phys. Rev. D 21, 1445, http://journals.aps.org/prd/abstract/10.1103/PhysRevD.21.1445 The Horowitz...
16. ### Grasping the Properties of Minkowski Space

I'm trying to get an intuitive feel for Minkowski space in the context of Special Relativity. I should mention that I have not studied (but hope to) the mathematics of topology, manifolds, curved spaced etc., but I'm loosely familiar with some of the basic concepts. I understand that spacetime...
17. ### Basis vectors of Minkowski space

Hi, I'm doing a first course in GR and have just found out that \eta_{ab} = g(\vec{e}_{a}, \vec{e}_{b}) = \vec{e}_{a} \cdot \vec{e}_{b} where g is a tensor, here taking the basis vectors of the space as arguments. I haven't seen this written explicitly anywhere but does this mean that...
18. ### Definition for curvature of worldline in Minkowski space

Hello, in this section of the wiki article on Rindler coordinates it is stated that the proper acceleration for an observer undergoing hyperbolic motion is just "the path curvature of the corresponding world line" and thus a nice analogy between the radii of a family of concentric circles and...
19. ### Understanding Minkowski Space Metrics: The Sign Reversal Mystery Explained

I've never seen a satisfactory explanation of the metrics used in a calculation of distance in Minkowski space. In Euclidean space, the distance is: ds^2 = dx^2 + dy^2 + dz^2 But in Minkowski space, the distance is ds^2 = (dt * c)^2 - dx^2 - dy^2 - dz^2 Why are the signs reversed? This implies...
20. ### Quick question on Minkowski space diagram

Consider the lower right plot in this picture (or many similar ones). I interpret the angle of the t' axis with respect to the t axis as: From the point of view of the stationary observer, all progress in time for the moving observer will be accompanied by the latter's spatial progress (to the...
21. ### Minkowski Space: Can Objects Have Same World Line?

Can two objects have same world line in Minkowski Space, if they are having same speeds ?
22. ### Doubled of Minkowski space and spinor wave function

First of all note that 8-dinensional Finsler space (t,x,y,z,t^*,x^*,y^*,z^*) preserving the metric form $$S^2 = tt^*-xx^*-yy^*-zz^*,$$ actually presents doubled of the Minkowski space. Then the solution with one-dimensional feature localized on the world line...
23. ### Circles in Minkowski space: unknown notation

I am reading an article about Minkowski space (as a vector space, which is why I am putting my question in this rubric) which is poorly translated from the Russian, and have come across several notational curiosities, most of which I have been able to figure out. However, there is one that I do...
24. ### Understanding Minkowski Space Compactification

Hi. I'm reading about the compactification of Minkowski Space, and there is a subject that is keeping me awake. They say that the group of conformal transformations is isomorphic to the group of pseudoorthogonal transformations with determinant equal to 1. I don't know how this happen and it...
25. ### What is the Definition of Continuity in Minkowski Space?

How "continuity" of a map Τ:M→M, where M is a Minkowski space, can be defined? Obviously I cannot use the "metric" induced by the minkowskian product: x\cdoty = -x^{0}y^{0}+x^{i}y^{i} for the definition of coninuity; it is a misinformer about the proximity of points. Should I use the Euclidean...
26. ### Hyperbolic path in Minkowski space

The path described by a constantly accelerating particle is given by: x=c\sqrt{c^2/a'^2+t^2} where a prime denotes an observer traveling with the particle and a letter without a prime a resting observer. If we leave the c^2/a'^2 out it reduces to x=ct, which makes sense. The distance...
27. ### Exploring Minkowski Space: Understanding Special Theory of Relativity

Hello friends, I am reading Einstein's special theory of relativity and came across this subject: Minkowski space. I cannot understand exactly what it is when i read that book and went to Wikipedia for understanding more about it. But i didn't understand much. What i...
28. ### 2D Systems and 4D Minkowski Space: Exploring Path Integrals

The combination of special relativity and quantum mechanics in a single framework makes our understanding of such systems to be true only in 4D, Minkowski space...I have noticed that recent published work concerning 2D systems and I am not sure about this reduction of 4D to only 2D, does it mean...
29. ### Fourier transform in Minkowski space

Hi, In Fourier analysis, we can decompose a function into sine waves with different wavenumbers that travel at different speeds (i.e., for a given wavenumber k they can have different frequencies ω and therefore different speeds v = ω/k). There is no upper bound on the speed of propagation v...
30. ### Geometric shape of Minkowski space

So, suppose for visualization there are only two dimensions: ct and x. Now if the metric where Euclidean, we could visualize this space is a simple plane. What would be the shape of the "plane" when the metric is +1, -1 (Minkowski)? Is it somehow hyperbolic?
31. ### Killing vectors of minkowski space

How does one know from the general form of the killing vectors in minkowski space: X^{a} = \omega_a_b(x^{a}) + t^{a} that there are 3 rotational isometries, 3 boosts, 3 spatial translations, and 1 time translation from that general form? It has me very confused >.<
32. ### Solving Einstein Field Equations for Minkowski Space with CTC

For Minkowski spacetime, the metric is: ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 I have read there is a solution when the time dimension is "rolled" into a cylinder forming a closed timelike curve. So the BC is t -> [0,T] with t = 0 identical with t = T. The Field Equation is: Rab - 1/2...
33. ### What is the different between Minkowski Space an Semi Riemann Space?

Last summer I took Semi Riemann Geometry lesson. Almost all the definitions in Semi Riemann geometry are with the same Minkowski geometry. I don't understand what is the different between Minkowski Space an Semi Riemann Space.
34. ### Local Minkowski space and free falling

Einstein's equivalence principle states that free-falling observers are in local inertial frame, so one can construct a local Minkowski frame everywhere. So my question is whether the logic can be inversed, does every local Minkowski space represent free-falling? because in vierbein...
35. ### Is Minkowski Space a Metric Space?: Exploring the Requirements and Implications

is minkowski space a metric space. As best as i can remember a metric space is a set with a metric that defines the open sets. With this intuition is Minkowski space a metric space. I mean i think it should be, but according to one of the requirements for a metric: d(x,y)=0 iff x=y triangle...
36. ### Is Minkowski space the only Poincare invariant space?

Hi everyone, I was wondering: if a space is invariant under Poincare transformations, does that mean it has to be Minkowski space? Or could it have some further isometries? By the same token, if a space is invariant under the orthogonal transformations, does it have to be Euclidean? I...
37. ### Minkowski Space Basics: Time, Light Speed & Purpose

Why is the vertical axis time multiplied by the speed of ligth? And, what is the purpose of using Minkowski space?
38. ### Would an energy diffraction ring in five space form a Minkowski space?

If you think of a sphereical symmetric diffraction ring, the intensity is constant for each sphereical section (intensity doesn't vary for theta or phi), but it varies kind of like a sine wave in the r dimension from zero to zero with a maximum in the middle of the ring. So that if you think...
39. ### Null curves vs. straight curves on Minkowski space

I understand that we could think of a null curve in Minkowski space as being the curve c(s) such that the tangent vector dc(s)/ds = 0 at all s. So suppose that we have a curve c(s) = (t(s), x(s), y(s), z(s)) and we want to ask ourselves what conditions would make c a straight line. I guess...
40. ### Is the Dual Basis in Minkowski Space Affected by the Metric Tensor?

Normally, if you have an orthonormal basis for a space, you can just apply your metric tensor to get your dual basis, since for an orthonormal basis all the dot products between the base vectors will boil down to a Kronecker delta. However, in Minkowski space, the dot product between a unit...
41. ### Compactness of V in Minkowski Space: A Question

Consider the closed forward light cone V = \left \lbrace x \in M \mid x^{2} \geq 0, x^{0} \geq 0 \right \rbrace and M denotes Minkowski space. My question is whether V is a compact set or not. If it is a compact set, how do I show it? Intuitively I would say it is compact, but I...
42. ### General Relativity and Minkowski space question

Is the curved spacetime of General Relativity possible to be described by Minkowskian space which is flat, or do one need some other geometry?
43. ### Structure of Minkowski Space

In another thread Fredrik referenced THE RICH STRUCTURE OF MINKOWSKI SPACE at http://arxiv.org/abs/0802.4345..(The math is NOT simple!) But the introduction got me wondering... around page three is this statement: So what knowledge regarding more exact models of spacetime and the relativity...
44. ### How can you create a geodesic path using the metric and initial velocity?

Let's say there is a small object heading towards Earth (it will burn up). It is first observed at: x^{\\mu}=[x^{1},x^{2},x^{2},x^{4}]=[x_{0},y_{0},z_{0},t_{0}] with a velocity: V_{v}=[v_{1},v_{2},v_{3},v_{4}] The metric is: ds^{2} = dx^{2} + dy^{2} + dz^{2} -c^{2}*dt^{2} g_{\\mu\\v} =...
45. ### Defining Velocity, Momentum, and Energy in Minkowski Space

I'm not completely sure were this post must be (math or here). But i got a question, I want to define velocity, momentum and energy. These looks like a simple task but let me explain the problem. I'm working in the Minkowski space, and the lorentz transformations (just geometrical one's)...
46. ### Vector Space Algebra of Minkowski Space

Consider the Minkowski space of 4 dimensions with signature (- + + +). How does the vector space algebra work here? More specifically given 3 space like orthonormal vectors how do we define fourth vector orthogonal to these vectors? I am looking for an appropriate vector product like it is in...
47. ### What is the Minkowski Metric and How Does it Relate to Special Relativity?

hey, i'm just trying to learn about special and general relativity and i figure a good place to start is with minkowski space since that is the basis of special relativity. I have a few questions though, i hope you forgive me because these questions will sound rather ignorant and silly i...
48. ### Relativity in Minkowski Space and Minkowski Diagram

hi :smile: I'm working on "Relativity on Minkowski Space and Minkowski Diagrams" as my undergraduate project. I have some references for my project but I want to make it perfect. May you introduce me some cool stuff (references, interestings, topics to work on & etc.) about this topic...
49. ### Fourier tr. of dirac delta in minkowski space

Hey everyone, a quick question: what is the Fourier space representation of the dirac delta function in minkowski space? It should be some integral over e^{ikx} (with some normalization with 2*pi's). I'm curious if the "kx" is a dot product in the minkowski or euclidean sense, and how one...
50. D

### Beginner Help on Minkowski Space (and others)

Please help me in any way you can. I'm a beginner physics student (entering undergraduate engineering) and currently self-taught. 1.) Minkowski space is illustrated in one of my books as a cone, with it's apex as the "origin", the side as the "light cone", the bend curvature inside the cone...