I'm trying to prove the following statements relating to space-like, time-like and light-like space-time intervals:(adsbygoogle = window.adsbygoogle || []).push({});

1.There exists a reference frame in which two space-time events aresimultaneousif and only if the two events arespace-likeseparated.

2.There exists a reference frame in which two space-time events arecoincident at a single spatial pointif and only if the two events aretime-likeseparated.

3.If two events arelight-likeseparated, then there arenoreference frames in which they aresimultaneousorcoincident at a single spatial point.

I can't seem to find any notes that a can verify my attempt with so I'm hoping that people won't mind taking a look at my workings on here to see if they're correct.

Consider the space-time interval [tex]\Delta S^{2}= (\Delta x^{0})^{2}-(\Delta\mathbf{x})^{2}[/tex] where we use the metric signature ##(+,-,-,-)##.

We then perform a spatial rotation of our coordinate system such that one of the space-time events ##x^{\mu}## is located at the origin of our reference, [tex]x^{\mu}=(0,0,0,0)[/tex] The other space-time event that we consider ##y^{\mu}## is thenspatiallyaligned along the ##z##-axis in our coordinate system, [tex]y^{\mu}=(y^{0},0,0,y^{3})[/tex]

Given this, we shall now consider each of the (numbered) cases above.

1. Space-like interval:

First, let the two events ##x^{\mu}## and ##y^{\mu}## besimultaneousin our inertial frame of reference ##S##, i.e. ##x^{0}=0=y^{0}##. It then follows that the space-time interval between them is given by [tex]\Delta S^{2}=(x^{\mu}-y^{\mu})^{2}=(0-0)^{2}-(0-y^{3})^{2}=-(y^{3})^{2}[/tex] Now, ##(y^{3})^{2}>0## and so clearly ##\Delta S^{2}<0##. Therefore, if the two events are simultaneous in $S$, then they arespace-likeseparated.

Next, let the two events ##x^{\mu}## and ##y^{\mu}## bespace-likeseparated. We have then that [tex]\Delta S^{2}=(x^{\mu}-y^{\mu})^{2}=(y^{0})^{2}-(y^{3})^{2}<0\quad\Rightarrow\quad\vert y^{0}\vert < \vert y^{3}\vert[/tex] Thus, we can choose ##\beta =v=\frac{y^{0}}{(y^{3})}## (in units where ##c=1##). From this, we see that ##\beta <1## as required. Performing a Lorentz boost along the ##z##-axis we can relate the coordinates ##x^{\mu}## and ##y^{\mu}## in ##S## to their expressions in another inertial frame ##S'## [tex]x'^{0}=\gamma\left(0-\beta 0\right)=0\; , \qquad x'^{3}=\gamma\left(0-\beta 0\right)=0[/tex] and [tex]y'^{0}=\gamma\left(y^{0}-\beta y^{3}\right)=0 \; , \qquad y'^{3}=\gamma\left(y^{3}-\beta y^{0}\right)[/tex] where ##\gamma =\frac{1}{\sqrt{1- \beta^{2}}}##.

Hence, in ##S'## we see that the two space-time events have the following coordinates [tex]x'^{\mu}=(0,0,0,0) , \qquad y'^{\mu}=(0,0,0,y'^{3})[/tex] and thus aresimultaneousin this frame.

2. Time-like interval:

This follows a very similar approach to the space-like case.

First, let the two events ##x^{\mu}## and ##y^{\mu}## becoincident at a single spatial pointin our inertial frame of reference ##S##, i.e. ##\mathbf{x}=\mathbf{0}=\mathbf{y}##. It then follows that the space-time interval between them is given by [tex]\Delta S^{2}=(x^{\mu}-y^{\mu})^{2}=(0-y^{0})^{2}-(0-0)^{2}=(y^{0})^{2}[/tex] Now, ##(y^{0})^{2}>0## and so clearly ##\Delta S^{2}>0##. Therefore, if the two events are spatially coincident in $S$, then they aretime-likeseparated.

Next, let the two events ##x^{\mu}## and ##y^{\mu}## betime-likeseparated. We have then that [tex]\Delta S^{2}=(x^{\mu}-y^{\mu})^{2}=(y^{0})^{2}-(y^{3})^{2}>0\quad\Rightarrow\quad\vert y^{0}\vert > \vert y^{3}\vert[/tex] Thus, we can choose ##\beta =v=\frac{y^{3}}{(y^{0})}## (in units where ##c=1##). From this, we see that ##\beta <1## as required. Performing a Lorentz boost along the ##z##-axis we can relate the coordinates ##x^{\mu}## and ##y^{\mu}## in ##S## to their expressions in another inertial frame ##S'## [tex]x'^{0}=\gamma\left(0-\beta 0\right)=0\; , \qquad x'^{3}=\gamma\left(0-\beta 0\right)=0[/tex] and [tex]y'^{0}=\gamma\left(y^{0}-\beta y^{3}\right) \; , \qquad y'^{3}=\gamma\left(y^{3}-\beta y^{0}\right)=0[/tex] where ##\gamma =\frac{1}{\sqrt{1-\beta^{2}}}##.

Hence, in ##S'## we see that the two space-time events have the following coordinates [tex]x'^{\mu}=(0,0,0,0)\; , \qquad y'^{\mu}=(y'^{0},0,0,0)[/tex] and thus arespatially coincidentin this frame.

3. Light-like interval:

In this last case it is trivial, as given two events ##x^{\mu}## and ##y^{\mu}##, if they arelight-likeseparated, then [tex]\Delta S^{2}=(x^{\mu}-y^{\mu})^{2}=(y^{0})^{2}-(y^{3})^{2}=0[/tex] and thus it is impossible to find a frame in which they are either simultaneous, or spatially coincident, as either one would change the interval into a space-like or a light-like interval. The interval is Lorentz invariant, so this clearly cannot be the case.

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Consequences of space-/time-/light-like separations

Loading...

Similar Threads - Consequences space light | Date |
---|---|

I How can General Relativity explain the Moon drifting apart from Earth | Mar 3, 2018 |

I What would be the consequences of following situation? | Mar 3, 2017 |

I Consequences of length contraction | May 20, 2016 |

B Einstein Train experiment with a consequence? | Mar 11, 2016 |

Trying to prove a consequence of harmonic gauge in GR | Aug 10, 2015 |

**Physics Forums - The Fusion of Science and Community**