Minkowski normal v. Euclidean normal

In summary, the difference between the normal in Minkowski spacetime and the normal in Euclidean space is that in Minkowski spacetime, some non-zero vectors are orthogonal to themselves, while in Euclidean space, the tangent line to a circle at a point P is perpendicular to the radius to that point.
  • #1
flyinjoe
17
0
What is the difference between the normal in Minkowski spacetime and the normal in Euclidean space?
 
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  • #3
Yes, sorry, I am sort of using normal as a synonym for orthogonal. Essentially, how would vector that is normal to a surface in Minkowski spacetime differ from a vector normal to the same surface embedded in Euclidean space?

I was thinking it had something to do with the magnitude function in the determination of the normal vector being different in each space (i.e., the denominator of the equation for a normal vector).

I'm sorry if this isn't making much sense.
 
  • #4
In a typical 1+1-dimensional spacetime diagram, a line through the origin with slope v is "orthogonal" (with respect to the Minkowski metric) to a line with slope 1/v. In particular, the x' axis in the image below is orthogonal to the t' axis.

500px-Minkowski_lightcone_lorentztransform.svg.png


In other words, as you tilt a line through the origin, the line through the origin that's orthogonal to it is tilted by the same amount as in the Euclidean case, but in the opposite direction.
 
  • #5
flyinjoe said:
What is the difference between the normal in Minkowski spacetime and the normal in Euclidean space?

In Minkowski space, some non-zero vectors are orthogonal to themselves, i.e., the hypersurface normal to a vector can include the vector!

When does this happen?
 
  • #6
The best definition for "normal" is the geometric one, as Minkowski defines in
en.wikisource.org/wiki/Translation:Space_and_Time
http://en.wikisource.org/wiki/Page:De_Raum_Zeit_Minkowski_018.jpg

In Euclidean geometry,
the tangent line to a circle at a point P is perpendicular to the radius to that point
(http://en.wikipedia.org/wiki/Tangent_lines_to_circles)

By analogy, in Minkowski spacetime,
the tangent line to a Minkowski-circle (a hyperbola) at a point P is Minkowski-perpendicular to the radius to that point.
This has an immediate physical intrepretation: an inertial observer's sense of "space [at an instant]" (what events are simultaneous for that observer) is perpendicular to that observer's sense of "time [along his worldline]".
 
  • #7
One good way of thinking about it is that when a timelike vector o is orthogonal to a spacelike vector s, an observer whose velocity four-vector is o considers s to be a vector of simultaneity. In other words, o points along that observer's time axis, and s could point along one of her spatial axes. (Of course this doesn't cover lightlike vectors or orthogonality of spacelike vectors to other spacelike vectors.) I have a treatment in this style in my SR book, http://www.lightandmatter.com/sr/ , at the beginning of the first chapter.
 
  • #8
flyinjoe...I'm not sure I really understand the answers so far...

maybe this will help a bit...

http://en.wikipedia.org/wiki/Minkowski_diagram

The first three sections, at a minium, provide insights.

You know that Minkowski is four dimensional and Eucledean is just the three of space, right?

So as I see it [simple minded perhaps] is that just as a normal to a plane has one less degree of freedom than a normal to a line embedded in that plane, so too is a normal restricted by the addition of a time component.

If you read between the lines in Wiki where it says

...Space and time have properties which lead to different rules for the translation of coordinates in case of moving observers [different than Newtonian/Eucledean]...In the Minkowski diagram this relativity of simultaneity corresponds with the introduction of a separate path axis for the moving observer.

and you can probably visualize different paths leads to different tangents and normals...
 

Related to Minkowski normal v. Euclidean normal

What is the difference between Minkowski normal and Euclidean normal?

Minkowski normal and Euclidean normal are two different ways of measuring distance in a mathematical space. Minkowski normal is used in a special type of space called Minkowski space, which is used in the theory of relativity. Euclidean normal, on the other hand, is used in traditional Euclidean space, which is the type of space we are most familiar with in everyday life.

Which space is more commonly used in scientific research?

Euclidean space is more commonly used in scientific research, as it is the type of space that most closely resembles our physical world. However, Minkowski space is also used in certain areas of physics, such as relativity and cosmology.

How do Minkowski normal and Euclidean normal differ in their measurement of distance?

In Euclidean space, distance is measured using the Pythagorean theorem, where the square of the distance between two points is equal to the sum of the squares of the differences in their coordinates. In Minkowski space, distance is measured using a different formula that takes into account the concept of spacetime and the speed of light.

Are there any real-world applications of Minkowski normal?

Yes, Minkowski space is used in the theory of relativity, which has numerous applications in modern technology, such as GPS systems and particle accelerators. It is also used in cosmology to understand the structure and evolution of the universe.

Which normal is more accurate in measuring distance?

Both Minkowski normal and Euclidean normal are accurate in measuring distance in their respective spaces. However, they are used in different contexts and cannot be directly compared in terms of accuracy.

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