28.2 Simultaneity And Time Dilation

Simultaneity and Relative Motion

In classical physics, time seems absolute: if two events happen at the same moment, every observer should agree on that. Special relativity overturns this assumption. Because the speed of light is the same in every reference frame, observers in relative motion can disagree about when events occur and even whether two events are simultaneous.

This section covers simultaneity, time dilation, and length contraction, three consequences of Einstein's postulates that become significant as speeds approach the speed of light.

Simultaneity and Relative Motion

Simultaneity in reference frames

Simultaneity means two events occur at the same instant within a given reference frame (an observer's perspective). The key insight of special relativity is that simultaneity is not absolute. Two events that are simultaneous for one observer may happen at different times for another observer moving relative to the first.

Why does this happen? Because the speed of light is constant in all inertial frames ($c \approx 3 \times 10^8 \text{ m/s}$). Light doesn't care whether you're moving toward it or away from it; it always reaches you at the same speed. This forces observers in relative motion to disagree about the timing of distant events.

A classic thought experiment illustrates this:

• A passenger stands in the middle of a moving train. Lightning strikes both ends of the train at the same instant (as measured by a person on the platform).
• The platform observer sees both flashes reach the midpoint at the same time, so the strikes are simultaneous for them.
• But the train passenger is moving toward the front flash and away from the rear flash. Since light travels at the same speed in both directions, the front flash reaches the passenger first. The passenger concludes the front strike happened before the rear strike.

Neither observer is wrong. Simultaneity genuinely depends on the reference frame. Einstein's synchronization method (using light signals and known distances) is how you establish simultaneity within a single frame, but it can't make two frames agree.

Time dilation and relative motion

Time dilation means that a clock moving relative to you ticks more slowly than your own clock. This isn't an illusion or a mechanical effect on the clock; it's a real feature of how time works at high speeds.

Time dilation follows directly from the constancy of light speed and the relativity of simultaneity. If two observers can't agree on simultaneity, they also can't agree on the elapsed time between events.

The effect is described by the Lorentz factor ($\gamma$):

$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$

where $v$ is the relative velocity between the two frames and $c$ is the speed of light.

A few things to notice about $\gamma$:

• At everyday speeds ($v \ll c$), $\gamma$ is essentially 1, so time dilation is negligible.
• At $v = 0.5c$, $\gamma \approx 1.15$, a 15% difference.
• At $v = 0.9c$, $\gamma \approx 2.29$, time passes at less than half the rate.
• As $v \to c$, $\gamma \to \infty$. You can never actually reach $c$ (for an object with mass), because it would require infinite energy.

A real-world example: cosmic ray muons. Muons are created high in the atmosphere and have a half-life of about $1.56 \, \mu s$. At that rate, most should decay before reaching Earth's surface. But they travel at roughly $0.98c$, giving $\gamma \approx 5$. From our perspective, their internal clocks run slow, so they survive long enough to be detected at ground level.

Time Dilation Calculations and Paradoxes

Calculations with the Lorentz factor

The core time dilation equation is:

$t = \gamma \, t_0$

where $t_0$ is the proper time and $t$ is the dilated time.

Here's a step-by-step example:

1. Identify the proper time. A spacecraft clock measures 10 years for a journey. Since the clock is at rest relative to the spacecraft, this is the proper time: $t_0 = 10 \text{ years}$.

2. Determine the relative velocity. The spacecraft travels at $v = 0.8c$ relative to Earth.

3. Calculate the Lorentz factor. $\gamma = \frac{1}{\sqrt{1 - \frac{(0.8c)^2}{c^2}}} = \frac{1}{\sqrt{1 - 0.64}} = \frac{1}{\sqrt{0.36}} = \frac{1}{0.6} \approx 1.667$

4. Find the dilated time. $t = \gamma \, t_0 = 1.667 \times 10 \text{ years} \approx 16.67 \text{ years}$

Earth clocks measure about 16.67 years for the same journey that the spacecraft clock measures as 10 years.

GPS satellites provide a practical example. They orbit at about 14,000 km/h, which is slow compared to $c$, but the onboard atomic clocks are precise enough that even tiny relativistic corrections (about 7 microseconds per day from special relativity alone) must be accounted for. Without these corrections, GPS positions would drift by kilometers.

Proper vs. dilated time

These two terms come up constantly, so make sure you can tell them apart:

• Proper time ($t_0$): The time interval measured by a clock that is at rest relative to the events being timed. The two events happen at the same location in this frame. Think of it as the time on your watch if the events happen right where you are.
• Dilated time ($t$): The time interval measured by a clock in a frame where the events occur at different locations. This is always longer than the proper time.

The relationship is always $t = \gamma \, t_0$, and since $\gamma \geq 1$, the dilated time is always greater than or equal to the proper time. Moving clocks run slow.

A common source of confusion: the astronaut on the spacecraft measures the proper time for events happening aboard the spacecraft, while mission control on Earth measures the dilated time for those same events. But if you flip the scenario and ask about events happening on Earth, then Earth's clock gives the proper time and the astronaut measures the dilated time. It always depends on where the events occur.

Twin paradox analysis

The twin paradox is the most famous thought experiment in special relativity. One twin stays on Earth while the other travels to a distant star at high speed and returns. When they reunite, the traveling twin has aged less.

Here's why it seems paradoxical: from the traveling twin's perspective, Earth is the one moving away and coming back. So shouldn't each twin see the other as younger? The situation looks symmetric.

The resolution comes down to acceleration:

1. The Earth twin stays in a single inertial (non-accelerating) frame the entire time.
2. The traveling twin must accelerate to leave, decelerate and turn around at the destination, and decelerate again upon return. These accelerations mean the traveling twin switches between different inertial frames.
3. This breaks the symmetry. Special relativity's time dilation formula applies cleanly to inertial frames, and only the Earth twin remains in one throughout.
4. A full analysis (using either general relativity or careful bookkeeping of frame changes) confirms that the traveling twin genuinely ages less. This is not a paradox at all; it's a consistent prediction.

With specific numbers: if the traveling twin moves at $0.8c$ and the Earth twin measures 20 years for the round trip, the traveling twin experiences only $20 / 1.667 \approx 12$ years.

Spacetime and Length Contraction

Spacetime concept

Special relativity unifies space and time into a single four-dimensional framework called spacetime. Every event is specified by four coordinates: three for position ($x, y, z$) and one for time ($t$).

Why combine them? Because space and time are no longer independent. How much time passes between two events depends on your motion through space, and vice versa. Different inertial observers "slice" spacetime differently, which is why they disagree about simultaneity and time intervals.

Inertial frames are reference frames moving at constant velocity relative to each other (no acceleration). The laws of physics, including the speed of light, are the same in all inertial frames. This is Einstein's first postulate.

Length contraction

Just as time intervals change between frames, so do spatial lengths. Length contraction means that an object moving relative to you appears shorter along the direction of motion than it does in its own rest frame.

The formula is:

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

where $L_0$ is the proper length (measured in the object's rest frame) and $L$ is the length measured by an observer relative to whom the object is moving.

Key points:

• Contraction happens only along the direction of motion. An object moving horizontally doesn't get shorter vertically.
• At $v = 0.9c$, $\gamma \approx 2.29$, so a 100-meter spacecraft would appear about 43.7 meters long to a stationary observer.
• Like time dilation, length contraction is negligible at everyday speeds and only becomes noticeable as $v$ approaches $c$.
• The proper length is always the longest measurement. Any observer seeing the object move will measure a shorter length.

Length contraction and time dilation are two sides of the same coin. In the muon example from earlier, you can explain the muon's survival either way: from Earth's frame, the muon's clock runs slow (time dilation); from the muon's frame, the distance to Earth's surface is contracted (length contraction). Both explanations give the same physical result.