What is the main difference between general relativity and special relativity?
Special relativity applies only to inertial frames and excludes gravity, while general relativity extends the principles to accelerated frames and includes gravitational effects via curved spacetime.
Einstein’s 1915 general theory redefined gravity in terms of the curvature of spacetime caused by mass and energy.
Explain the implications of the postulate that states the speed of light in a vacuum is constant for all observers, regardless of their motion relative to the light source.
- The postulate directly leads to the Lorentz transformations.
- These imply the existence of spacetime, so that space and time are fundamentally linked with each other.
- This replaces classical Galilean transformations.
The constancy of light speed implies that time and space must adjust to preserve this invariant speed across all inertial frames. This leads to observable effects such as time dilation, length contraction, and the relativity of simultaneity.
It also ensures that Maxwell’s equations (and all physical laws) remain valid in all inertial reference frames, resolving contradictions between classical mechanics and electromagnetism and laying the foundation for special relativity.
Explain how simultaneity is affected by the relative motion of observers according to the theory of special relativity.
In special relativity, simultaneity is relative and depends on the observer’s frame of reference.
Two events that are simultaneous in one inertial frame may not be simultaneous in another frame moving relative to the first. Suppose that we have two events in one frame that are spatially separated. Then Δt = 0 while Δx is not. It follows immediately from Δt′ = γ · (Δt − v Δx / c²) that Δt′ is not zero.
True or False:
If, in one frame, two events take place at the same location and are simultaneous, they are simultaneous in every other inertial reference frame.
True
This follows from Δt′ = γ · (Δt − v Δx / c²) with Δx = 0 and Δt = 0.
Write down the Lorentz transformations.
- x′ = γ (x − vt)
- t′ = γ (t − vx / c²)
- y′ = y
- z′ = z
Remember that γ = 1 / √(1 − v² / c²).
We have set up axes such that motion is taking place along the x-axis. The inverse transformation are obtained simply by swapping the primed and unprimed quantities, and flipping the sign of v.
True or False:
In special relativity, the spacetime interval is invariant. (assuming motion is taking place along the x-axis).
True
This follows from the Lorentz transformations for intervals.
- Δx′ = γ · (Δx − v Δt)
- Δt′ = γ · (Δt − v Δx / c²)
Simple algebra leads to (Δs’)2 = c2 (Δt)2 - (Δx)2 = (Δs)2.
Due to the minus sign, this means that the geometry of spacetime is non-Euclidean.
Fill in the blank:
The relativistic energy-momentum relation is expressed as ______, where ( E ) is the total energy, ( p ) is the momentum, and ( m ) is the mass.
This relation underscores the equivalence of mass and energy and provides a means to calculate the energy of a particle moving at relativistic speeds.
Consider an observer with a stationary (in his frame) clock. The time interval between two ticks of the clock as measured by this observer is Δt.
What is this time interval as measured by a second observer moving with velocity v with respect to the first observer?
- This is time dilation.
- The time interval between the ticks of the clock as measured by the first observer is the proper time. The two events (the first tick and the second tick) take place at the same location for this observer. So Δx = 0.
- It follows then from the Lorentz transformations that Δt’ = γΔt.
- Since γ > 1 (remember that γ depends on v), we have time dilation: moving clocks run slow.
Time dilation has been confirmed by numerous experiments, such as those involving high-speed particles and atomic clocks in jets.
True or False:
Time dilation effects become negligible at velocities much less than the speed of light.
True
At speeds much slower than the speed of light, ( v ≪ c ), the Lorentz factor γ is approximately equal to one, meaning time dilation is not significant.
An astronaut travels to a star system 5 light-years away (as measured in Earth’s frame) at a constant speed of 0.6c.
According to the astronaut’s clock, how much time elapses during the trip?
There are two events: the astronaut leaves the Earth, and the astronaut arrives at the star system. Both events take place at the same location in the reference frame of the astronaut. So the astronaut measures the proper time.
In the reference frame of the Earth, the time interval is Δt’ = (5 ly)/(0.6c).
Fill in the blank:
As velocity approaches the speed of light the Lorentz factor approaches ______.
Assume motion in an inertial frame and speeds approaching c, the speed of light in a vacuum.
infinity
As v approaches the speed of light, the denominator approaches zero, causing γ to approach infinity.
The Lorentz factor, also called the time dilation factor, increases rapidly as velocity approaches the speed of light, illustrating how relativistic effects dominate at high speeds.
Analyze the implications of time dilation for high-speed particle decay experiments.
- Particles moving at relativistic speeds exhibit longer decay times from the lab frame perspective.
- Experimental confirmation of time dilation: muon decay in the atmosphere and particle accelerators.
- Necessitates relativistic corrections in experimental physics.
Time dilation is crucial for accurately describing the behavior of fast-moving particles, aligning experimental observations with theoretical predictions. When the value of the lifetime of a particle is quoted, it is understood that this is the time taken (on average) by the particle to decay in its rest frame. The time taken by the particle to decay in the lab frame will be longer (how much longer depends on the speed of the particle).
Consider two twins. One of them remains on Earth, while the other travels on a spaceship to a distant planet and then comes back to Earth. According to the twin on Earth, the age of the traveling twin is now smaller due to time dilation.
However, if we analyze the situation in the frame of the traveling twin, it is the twin on Earth who was moving, so the age of the twin on Earth should be smaller.
We have a paradox. What is the resolution?
The traveling twin is younger.
This conclusion is clear if we analyze the situation in the Earth frame. We cannot analyze the situation from the viewpoint of the traveling twin in a straightforward manner - even if the speed of the spaceship is constant, this twin is in a non-inertial frame since the spaceship reverses direction.
To analyze the situation from the traveling twin’s perspective, we must consider two separate inertial frames: one for the outbound journey and another for the return. The traveling twin transitions from the first frame to the second. If we take this transition into account, we do find that the traveling twin is younger.
The twin paradox highlights the non-intuitive consequences of time dilation and the role of inertial vs. non-inertial frames in special relativity.
Maria and Anna are twins and are 30 years old. Maria sets out on a round trip to the star Sirius, located 8.7 light years from Earth, at a speed of 0.95 c. What are the twins’ ages upon reunion?
- Ana is 48.3 years
- Maria is 35.7 years
- The total round trip takes (2 x 8.7)/0.95 = 18.32 years in the Earth frame, so Ana’s age is 48.3 years.
- The journey from Earth to Sirius takes 9.16 years in the Earth frame.
- The two events (Maria leaving Earth and Maria arriving at Sirius) occur at the same location for Maria (so Maria measures the proper time).
- The time taken by Maria to reach Sirius then is 9.16/γ = 2.86 years.
- The total time taken for Maria is 5.72 years. So Maria’s age will be 35.7 years.
The half-life of a pion moving at high speed turns out to be 60 ns, while its half-life at rest is 26 ns.
What is its speed?
v ≈ 0.9c
Use τ=τ₀/√(1-v²/c²) to solve for v. The proper time is 26 ns.
What is length contraction?
The phenomenon in special relativity where the length of an object measured in a frame where it is moving is shorter than its proper length L0 (the length in its rest frame).
The relationship is given by L = L₀ / γ, where γ = 1 / √(1 − v² / c²) is the Lorentz factor, and v is the relative velocity between the observer and the object.
True or False:
Consider a stationary observer on a train station and another observer in a moving train (moving with respect to the observer on the train station).
According to the observer on the train station, objects in the train are Lorentz contracted, while according to observer on the train, object on the train station are Lorentz contracted.
Assume both observers are in inertial frames and applying special relativity.
True
This follows from the first postulate of special relativity that says that all inertial reference frames are equivalent. There is no paradox here (although it may seem so at first glance). For an object’s length to be measured, the positions of its ends need to be measured at the same time.
When such a measurement is done by one observer, the other observer thinks that the positions were not measured at the same time, and vice versa.
Fill in the blanks:
In the context of length contraction, the length measured in the rest frame of the object is known as the ______ ______.
Proper length
The proper length is always the longest measurement of an object’s length, as it is measured in the frame where the object is at rest.
Discuss the implications of length contraction on the concept of rigid bodies in classical physics.
- In classical physics, a rigid body is an object whose shape and size do not change, regardless of motion or applied forces.
- Length contraction implies that no perfectly rigid bodies exist in relativity, as objects deform under motion.
- The length parallel to the motion gets shortened, while the transverse lengths do not.
- This length contraction is frame-dependent - different observers moving at different velocities will measure different shapes of the object.
- Consequently, in relativistic physics, since perfect rigidity is impossible, bodies are modeled as elastic or deformable, with internal forces propagating at finite speeds.
These insights lead to a deeper understanding of material properties in relativistic contexts.
Two identical spaceships, connected by a string, are initially at rest in an inertial frame. They then accelerate identically in this frame.
Will the string break?
Yes
Their separation remains constant in this frame (where they are initially at rest) since at any instant of time, they have the same velocity. However, the string is Lorentz contracted. This means that the string will eventually break.
On the other hand, in the frame where the spaceships are momentarily at rest, the distance between the spaceships increases because the accelerations of the spaceships are not simultaneous. Therefore, according to this frame too, the string eventually breaks.
This is known as the Bell spaceship paradox.
A ladder of proper length longer than a garage is moving (in the inertial frame where the garage is at rest) so fast that the Lorentz contracted length is smaller than the length of the garage.
The ladder then appears to fit inside the garage. But in the ladder’s frame, it is the garage that is shorter.
What resolves this apparent paradox?
Assume both garage and ladder are inertial and rigid, and that we are applying special relativity only.
There is no paradox.
In the rest frame of the garage, the ladder is Lorentz contracted and therefore fits entirely within the garage at a single instant—both ends are simultaneously inside. However, in the rest frame of the ladder, it is the garage that is contracted and appears too short to contain the ladder.
The resolution lies in the relativity of simultaneity: events that are simultaneous in one frame are not necessarily simultaneous in another. In the ladder’s frame, the front and back of the ladder are not inside the garage at the same time. Thus, there is no contradiction—both perspectives are consistent within the framework of special relativity.
Simultaneity is frame-dependent, resolving the paradox.
Why did Maxwell’s equations lead to the need for special relativity?
Maxwell’s equations change form under Galilean transformations. This meant that the laws of electromagnetism are not the same for all inertial frames (under Galilean transformations).
In particular, the speed of light (whose value can be found from Maxwell’s equation) would be different in different inertial frames - perhaps there was a special frame in which the speed of light was equal to the value predicted by Maxwell’s equation (the ether frame).
Einstein proposed that this cannot be so - all physical laws are the same for all physical observables. Space and time transform according to Lorentz transformations (which reduce to Galilean transformations at low speeds), and under Lorentz transformations, Maxwell’s equations are the same for different inertial frames.
The failure to detect the ether in experiments like the Michelson-Morley experiment reinforced this shift.
Derive the condition under which two events are simultaneous for an observer moving with velocity ( v ) relative to another observer.
For two events to be simultaneous in the moving observer’s frame, the time difference Δ t’ between the events must be zero.
Using Lorentz transformations:
Δt’ = γ( Δt - (v Δx)/(c^2) = 0
Solving for Δt, we find:
Δt = (v Δx)/(c^2)
Two events that are not simultaneous in one frame can be simultaneous in another frame.
True or False:
For two events such that (Δs)2 = (cΔt)2 - (Δx)2 > 0, the temporal order of the two events depends on the reference frame.
False
If the spacetime interval between two events is timelike, that means they can be causally connected, one could potentially influence the other. Now, if one observer sees the first event happen before the second, the question is whether another observer (in a different inertial frame) could see the reverse order.
Using the Lorentz transformation, we find that reversing the time order would require a relative speed faster than the speed of light. Since that’s impossible, the order of timelike events is the same in all reference frames, causality is preserved.
For spacelike separated events where (Δs)2 < 0), their order can appear different to different observers.
