Timelike Metrics: What Is a Timelike Vector?

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In summary, the given metric of Scwarzchild has a coefficient in front of the dxdx term that is <0 for timelike vectors and >0 for spacelike vectors. A timelike vector is a killing vector that makes all the metric components vanish, and can be determined by the sign of gtt. In general, a vector is considered time-like at a point if gtt is negative based on the -+++ convention.
  • #1
pleasehelpmeno
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In a given metric say Scwarzchild is [itex]\frac{\partial}{\partial t}[/itex] time-like when the coefficient in front of the dxdx term is <0 and space-like when the coefficients in front of spatial terms >0 ?

and what is a timelike vector is it simply a vector in the coefficient that satisfies the above criteria?
 
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  • #2
pleasehelpmeno said:
In a given metric say Scwarzchild is [itex]\frac{\partial}{\partial t}[/itex] time-like when the coefficient in front of the dxdx term is <0 and space-like when the coefficients in front of spatial terms >0 ?

and what is a timelike vector is it simply a vector in the coefficient that satisfies the above criteria?

Are you talking about the killing vector? That's the abbreviated notation normally used for it. You see that it is a killing vector in this case because it makes all the metric components vanish. You see that it is timelike or spacelike by the sign of gtt.
 
  • #3
PAllen said:
Are you talking about the killing vector? That's the abbreviated notation normally used for it. You see that it is a killing vector in this case because it makes all the metric components vanish. You see that it is timelike or spacelike by the sign of gtt.
I believe he is talking about the basis vector that is dual to [itex]dt[/itex] i.e. [itex]dt(\partial t) = 1[/itex]. This is very coordinate chart specific but yes as PAllen already stated it is based on the sign of gtt. In general on a space - time M, at [itex]p\in M[/itex] some [itex]v\in T_{p}(M)[/itex] is time - like if [itex]g_{p}(v,v) < 0[/itex] using the -+++ convention.
 

Related to Timelike Metrics: What Is a Timelike Vector?

1. What is a timelike vector in the context of timelike metrics?

A timelike vector is a mathematical concept used in the study of spacetime in general relativity. It is a vector that, when multiplied by itself, results in a negative value. In the context of timelike metrics, timelike vectors represent the time direction in spacetime and are used to describe the movement of particles through time.

2. How is a timelike vector different from a spacelike vector?

A timelike vector and a spacelike vector are opposites in terms of their mathematical properties. While a timelike vector has a negative length when multiplied by itself, a spacelike vector has a positive length. In the context of timelike metrics, spacelike vectors represent the spatial directions in spacetime and are used to describe the movement of particles through space.

3. What is the significance of timelike metrics in general relativity?

In general relativity, timelike metrics are used to describe the curvature of spacetime caused by the presence of massive objects. They allow us to calculate the paths of particles through spacetime, taking into account the effects of gravity. Timelike metrics are essential for understanding the behavior of matter and energy in the universe.

4. Can an object move in both a timelike and spacelike direction simultaneously?

No, an object can only move in one direction at a time in the context of timelike metrics. This is because timelike and spacelike vectors are mutually exclusive and cannot be combined. An object can move in a combination of timelike and spacelike directions if it is described by a null vector, which has a length of zero when multiplied by itself.

5. Are there any real-world applications of timelike metrics?

Yes, timelike metrics have many real-world applications, particularly in the fields of astrophysics and cosmology. They are used to study the behavior of objects in space, including planets, stars, and galaxies. They also play a crucial role in the development of technologies such as GPS and satellite communications, which rely on the accurate measurement of time and space.

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