A.5.3 Lorentz transformations (HL only)

Lorentz Transformations

They ensure that the speed of light remains constant in all inertial frames, a key postulate of special relativity.

Deriving the Lorentz Transformations

Consider two inertial frames, $S$ and $S^{\prime }$:

• $S^{\prime }$ moves with velocity $v$ relative to $c$ along the x-axis.
• The origins of $S$ and $S^{\prime }$ coincide at $t=t^{\prime }=0$.

Transformation for Position

• In Galilean relativity, the position transformation is:

$$x^{\prime }=x-vt$$

• However, this does not account for the constancy of the speed of light.
• The Lorentz transformation modifies this to:

$$x^{\prime }=\gamma (x-vt)$$

where $$\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$ is the Lorentz factor.

Transformation for Time

1. In Galilean relativity, time is absolute ($t^{\prime }=t$).
2. However, in special relativity, time is relative.
3. The Lorentz transformation for time is:

$$t^{\prime }=\gamma (t-\frac{vx}{c^2})$$

Why the Lorentz Factor?

1. The Lorentz factor $\gamma$ ensures that the speed of light is constant in all inertial frames.
2. It approaches 1 at low speeds (recovering Galilean transformations) and increases significantly as $v$ approaches $c$.

Time Dilation

Deriving the Time Dilation Formula

1. Consider a clock at rest in frame $S^{\prime }$.
2. It ticks twice, separated by a proper time interval $\mathrm{\Delta }{t}_0$ (measured in the clock’s rest frame).
3. An observer in frame $S$ sees the clock moving with velocity $v$.

Using the Lorentz transformation for time:

$$t^{\prime }=\gamma (t-\frac{vx}{c^2})$$

Since the clock is at rest in $S^{\prime }$, $\mathrm{\Delta }{x}^{\prime }=0$. Therefore:

$$\mathrm{\Delta }t=\gamma \mathrm{\Delta }{t}_0$$

Relativity of Simultaneity

• In special relativity, the concept of simultaneity is not absolute; events that are simultaneous in one inertial frame may not be simultaneous in another moving frame.

1. Consider two events occurring at different positions, $x_1$ and $x_2$, in frame $S$ but at the same time, $t_1=t_2$.
2. An observer in another frame, $S^{\prime }$, moving with velocity $v$ relative to $S$, will measure the time difference between the events as: $$\mathrm{\Delta }{t}^{\prime }=\gamma (\mathrm{\Delta }t-\frac{v\mathrm{\Delta }x}{c^2})$$
3. Since $\mathrm{\Delta }t=0$ in the original frame ($S$), the time interval observed in $S^{\prime }$ becomes: $$\mathrm{\Delta }{t}^{\prime }=-\gamma \frac{v\mathrm{\Delta }x}{c^2}$$

• That equation shows that the time difference between events depends on their spatial separation ($\mathrm{\Delta }x$) and the relative velocity ($v$) of the frames.
• Thus, simultaneity is relative and depends on the observer's frame of reference.

Spacetime Interval: An Invariant Quantity

1. In special relativity, the spacetime interval ($\mathrm{\Delta }s$) between two events remains invariant across all inertial reference frames.
2. This means that even though different observers may measure different distances and time intervals, the spacetime interval is the same for all observers.
3. It is mathematically expressed as:

$$(\mathrm{\Delta }s{)}^2=(c\mathrm{\Delta }t{)}^2-(\mathrm{\Delta }x{)}^2$$

where:

• $\mathrm{\Delta }s$ is the spacetime interval,
• $c\mathrm{\Delta }t$ represents the separation in time (converted into distance units by multiplying by $c$),
• $\mathrm{\Delta }x$ represents the spatial separation between the events.

Why is the Spacetime Interval Invariant?

• The invariance of the spacetime interval means that all inertial observers—regardless of their motion—agree on its value, even though they may disagree on $\mathrm{\Delta }x$ and $\mathrm{\Delta }t$.

Interpretation and Consequences

• If $(\mathrm{\Delta }s{)}^2>0$, the interval is timelike:
  • $⟹$ The two events are causally connected, and a signal traveling at or below the speed of light could travel between them.
• If $(\mathrm{\Delta }s{)}^2=0$, the interval is lightlike:
  • $⟹$ The events are connected by a signal traveling exactly at the speed of light.
• If $(\mathrm{\Delta }s{)}^2<0$, the interval is spacelike:
  • $⟹$ No signal, not even light, could travel between the events, making them causally disconnected.

Length Contraction

Deriving the Length Contraction Formula

1. Consider a rod at rest in frame $S^{\prime }$. Its proper length (measured in its rest frame) is $L_0$.
2. An observer in frame $S$ measures the length of the rod while it moves with velocity $v$.

Using the Lorentz transformation for position:

$$x^{\prime }=\gamma (x-vt)$$

The length measured in $S$ is:

$$L=x_2-x_1$$

The proper length is:

$$L_0=x_2^{\prime }-x_1^{\prime }=\gamma (x_2-v{t}_2)-\gamma (x_1-v{t}_1)$$

Since the measurement is simultaneous in $S$ ($t_2=t_1$), the equation simplifies to:

$$L_0=\gamma L$$

Solving for $L$ gives the length contraction formula:

$$L=\frac{L_0}{\gamma }$$

Relativistic Velocity Addition

1. At relativistic speeds, the classical formula for adding velocities ($u=u^{\prime }+v$) breaks down.
2. Instead, the relativistic velocity addition formula is used:

$$u=\frac{u^{\prime }+v}{1+\frac{u^{\prime }v}{c^2}}$$

Reflection

The consequences of the constancy of the speed of light challenge our intuitive understanding of space and time.