Timelike and lightlike/null vectors

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In summary, the conversation discusses the problem of showing that a timelike vector cannot be orthogonal to a null vector. It presents equations and arguments involving the line element of Minkowski space-time and the components of the vectors. Ultimately, it concludes that the inequality of the components of the vectors contradicts the condition for orthogonality.
  • #1
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I have another problem that I am stuck on.

Show that a timelike vector cannot be orthogonal to a null vector.

Timelike:
X[tex]^{2}[/tex] = g(sub a b)X[tex]^{a}[/tex]X[tex]^{b}[/tex] > 0

Null:
X[tex]^{2}[/tex] = g(sub a b)X[tex]^{a}[/tex]X[tex]^{b}[/tex] = 0

In order for them to be orthogonal...

g(sub a b)X[tex]^{a}[/tex]Y[tex]^{a}[/tex] = 0

I know from the line element of Minkowski space-time you have

ds[tex]^{2}[/tex] = dt[tex]^{2}[/tex] - dx[tex]^{2}[/tex] - dy[tex]^{2}[/tex] - dz[tex]^{2}[/tex]

I have played around with substituting the line element into the equation that specifies if it is orthogonal, but to no success. Any help would be great!

Sorry for the bad writting, I couldn't get latex to do subscripts, it only did them in superscripts...

Thanks,

Chris
 
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  • #2
Here's a possible argument. Treat each 4 vector as e.g. (x0, r) where r is a 3 vector, x0 the timelike component. Let Y be a null vector, X be timelike vector. Then we have:

(1) Y0^2 - ry dot ry = 0, where dot is euclidean 3 dot product.

(2) X0 Y0 - rx dot ry = 0

(3) X0^2 - rx dot rx > 0

From which:

x0 > norm(rx) , 3 norm, euclidean, by (3)

Y0 = norm (ry) by (1)

X0 Y0 > norm(rx) norm(ry) >= rx dot ry

which contradicts (2).
 
  • #3
thanks =)
 

1. What is the difference between timelike and lightlike/null vectors?

Timelike vectors are those that have a positive length (or squared length) and are associated with objects that move slower than the speed of light. Lightlike/null vectors, on the other hand, have a length of zero and are associated with objects that move at the speed of light.

2. How are timelike and lightlike/null vectors used in physics?

Timelike and lightlike/null vectors are used in the study of spacetime in physics, particularly in the theory of relativity. They help describe the relationships between time and space, as well as the speed and motion of objects in the universe.

3. Can timelike or lightlike/null vectors be negative?

No, timelike and lightlike/null vectors cannot be negative. They are defined by their length or squared length, which is always a positive value. Negative values would not make sense in the context of these vectors.

4. How is the concept of causality related to timelike and lightlike/null vectors?

In physics, causality refers to the relationship between cause and effect. Timelike vectors are associated with events that can causally affect each other, while lightlike/null vectors are associated with events that cannot causally affect each other. This is because objects moving at the speed of light cannot interact with each other in a cause-and-effect manner.

5. Are there any real-world applications of timelike and lightlike/null vectors?

Yes, timelike and lightlike/null vectors have many real-world applications. They are used in the development of GPS technology, the study of black holes and space-time curvature, and in the prediction of astronomical events such as eclipses. They are also important in the study of particle physics and the behavior of subatomic particles.

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