# Four-vectors, Minkowsky spacetime.

• norbert
In summary, the terms "timelike," "spacelike," and "lightlike" are used in relativistic electrodynamic to describe different possibilities for the spacetime distance between two events. Timelike separation means that the events are chronologically connected, while spacelike separation means that they are causally connected. Lightlike separation means that a photon can pass between the events. These terms are based on the signature of the metric being used and can be interpreted in terms of inertial observers and the relationship between time and space in the given context.
norbert
they often are used in relativistic electrodynamic
i.e when I´m referring to the lightcone in a spacetime diagram.
I would like some physics-interpretation of this concepts.
thanks.
D. Norbert

These basically apply to three different possibilities for the spacetime distance between two events: positive, negative or zero. Lightlike (or null separated) events have zero distance between them; how timelike and spacelike match up with positive and negative depends on the signature of your metric. Suppose we use Minkowski metric $\eta_{\alpha\beta}=diag(1,-1,-1,-1)$, then spacelike means $ds^2<0$ and timelike means $ds^2>0$. Here's where the names come from.

If we have two events such that the spatial distance between them is equal to the time between them times the speed of light, i.e. $ds^2=c^2dt^2-dx^2-dy^2-dz^2=0$ then the events are lightlike since they can be connected by the path of a photon.

If there are two events that can't be joined by the path of any real particle (i.e. one traveling at or below the speed of light) then the events are spacelike. An example is two simultaneous (in a given frame) events at different spatial locations. Then $dt^2=0$ since no time elapses between the events, and $ds^2=-dx^2-dy^2-dz^2<0$. You can sort of think of it as being that they are spacelike because there is more space than time between them'.

Finally, two events which can be joined by the path of a massive particle (i.e. traveling with v<c) are timelike - e.g. more time than space between them' such as two events that happen in the same place but at different times. Then there is no spatial distance between them so $dx^2=dy^2=dz^2=0$ and we have $ds^2=c^2dt^2>0$.

Just to add to mikeu's post,
timelike is a direction from the vertex pointing inside the lightcone,
lightlike (or null) is a direction from the vertex pointing along the lightcone,
spacelike is a direction from the vertex pointing outside the lightcone.

These adjectives apply to the generic four-vector, as well as the displacement four-vector.

Physically:
Timelike-related events are said to be "chronologically connected".
Nonspacelike-related events are said to be "causally connected".
Spacelike-related events are not causally connected. Such events cannot influence each other.

The physical interpretation is:

• If two events have a timelike separation, then there is an inertial observer who thinks they occur at the same place
• If two events have a spacelike separation, then there is an inertial observer who thinks they occur at the same time
• If two events have a null (lightlike) separation, then a photon can pass between them

## 1. What is a four-vector?

A four-vector is a mathematical concept used in physics to represent a physical quantity that has both magnitude and direction in four-dimensional spacetime. It is typically denoted by a four-component vector, with each component representing a different dimension (three for space and one for time).

## 2. What is Minkowsky spacetime?

Minkowsky spacetime is a mathematical model of the four-dimensional spacetime that is used in special relativity. It combines the three dimensions of space with the dimension of time to create a four-dimensional continuum, where the laws of physics are the same for all observers in uniform motion.

## 3. How are four-vectors and Minkowsky spacetime related?

Four-vectors are used to describe physical quantities in Minkowsky spacetime. The components of a four-vector represent the position, velocity, momentum, or any other physical quantity of an object in spacetime. The geometry of Minkowsky spacetime allows for the use of four-vectors to accurately describe the properties of objects in motion.

## 4. What is the significance of the Minkowsky metric?

The Minkowsky metric is a mathematical tool used to calculate the distance or interval between two events in Minkowsky spacetime. It is a fundamental concept in special relativity, as it allows for the determination of the spacetime interval, which is an invariant quantity that remains the same for all observers in uniform motion.

## 5. How is Minkowsky spacetime different from traditional Euclidean spacetime?

Minkowsky spacetime differs from traditional Euclidean spacetime in that it includes the dimension of time, which means that distances and intervals are measured differently. In Euclidean spacetime, distances are measured using the Pythagorean theorem, while in Minkowsky spacetime, the Minkowsky metric is used. Additionally, the laws of physics are different in the two models, as Minkowsky spacetime incorporates the principles of special relativity.

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