A.5 Galilean and special relativity (HL only)

Einstein's special relativity is built on exactly two postulates. The first concerns the universality of physical laws across inertial frames; the second concerns the constancy of the speed of light.

The first postulate states that the laws of physics are identical in all inertial (non-accelerating) frames. It does NOT extend to accelerating frames, which are handled by general relativity.

The second postulate states that $c\approx 3\times 10^8{\text{ m s}}^{-1}$ is the same for all inertial observers, completely independent of the motion of either the source or the observer. This is the radical departure from Galilean mechanics.

The option that correctly captures both postulates — inertial frames only, and $c$ independent of source/observer motion — is the second option.

We need to identify the option that correctly states both postulates without any errors or omissions.

Option 1 states that the speed of light depends on the velocity of the source — this is exactly what special relativity disproves. This option contradicts the second postulate entirely.

Option 3 claims time is absolute for all observers. This is the Newtonian assumption that special relativity explicitly overthrows. It contradicts the fundamental conclusions of the theory.

Option 4 incorrectly restricts the constancy of $c$ to measurements by a stationary observer. The entire point of the second postulate is that $c$ is constant for all inertial observers, moving or not.

Only option 2 correctly states both postulates: laws of physics apply in all inertial frames, and $c$ is constant for all inertial observers regardless of source or observer motion.

The Lorentz factor is defined as: $\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

$\gamma =\frac{1}{\sqrt{1-(0.6{)}^2}}=\frac{1}{\sqrt{1-0.36}}=\frac{1}{\sqrt{0.64}}$

$\sqrt{0.64}=0.8$ $\gamma =\frac{1}{0.8}=1.25$

Since $\gamma \ge 1$ always and the object is moving, $\gamma =1.25$ is physically consistent. The correct answer is $\gamma =1.25$.

We must compute $\gamma$ for $v=0.6c$ and select the correct value from the options.

The Lorentz factor is always greater than or equal to 1. A value of $0.8<1$ is physically impossible for any real velocity. This option is eliminated immediately.

The value $1.67\approx 5/3$ corresponds to $v=0.8c$ (since $\sqrt{1-0.64}=0.6$, giving $\gamma =5/3$), not $v=0.6c$. This is a common mix-up between $0.6c$ and $0.8c$.

$\gamma =2$ corresponds to $v=0.866c$, since $\sqrt{1-0.75}=0.5$. This is much higher than $0.6c$, so this option is incorrect.

Calculating directly: $\gamma =1/\sqrt{1-0.36}=1/0.8=1.25$. The correct answer is $\gamma =1.25$.