# Four-vectors, Minkowsky spacetime.

1. Sep 3, 2005

### 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

2. Sep 3, 2005

### mikeu

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 travelling 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. travelling 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 $ds^2=c^2dt^2>0$.

3. Sep 3, 2005

### robphy

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.

4. Sep 5, 2005

### DrGreg

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