A street performer tosses a ball straight up into the air...

A street performer tosses a ball straight up into the air (event 1 ) and then catches it in his mouth (event 2 ). For each of the following observers, state whether the time they measure between these two events is the proper time or the dilated time: (a) the street performer; (b) a stationary observer on the other side of the street; $(c)$ a person sitting at home watching the performance on $T V ;$ (d) a person observing the performance from a moving car.

Key Concepts

Time Dilation

Time dilation is a consequence of special relativity stating that the time interval between two events will appear longer to an observer who sees these events occurring at different spatial locations (or when there is relative motion between the observer and the events). This results from the Lorentz transformation of time when switching between different inertial frames.

Reference Frame

A reference frame is a coordinate system used by an observer to measure positions and times of events. In the context of relativity, the measurements of time (such as proper time and dilated time) depend on the observer's state of motion relative to the events, making the choice of reference frame crucial to the analysis.

Proper Time

Proper time is the time interval between two events as measured by an observer for whom the events occur at the same location in space. It is considered the 'true' elapsed time experienced by an observer moving along the path connecting the events, and it always represents the minimum possible time interval between them.

Transcript

00:02 Hello, and in this question here, i'm going to explain the difference between proper and dilated time.

00:09 So let's consider a moving light bulb, okay? and this light bulb, okay, travels from a to b, with a constant velocity that is equal to v.

00:29 Okay? at position a, it's going to flash once the light bulb, and we call this event one.

00:39 At position b, it's going to flash again and we call this event two.

00:46 So how do these events look to different observers? well, let's say we have an observer, okay, who is travelling with the same velocity as the light bulb.

01:00 Okay, so he's traveling with a velocity v with the light bulb.

01:07 So to him, it looks like event one and event two, a car in the exact same position in space.

01:16 Okay? and this is because there is no relative velocity between him and the light bulb as they are both moving at the same velocity.

01:30 So observer 1 who moves at v.

01:42 Both events, events occurring in the same spatial position.

02:05 Okay? and because he sees both events occurring in the same spatial position, we call the time he measures or she measures to be the proper time.

02:30 By contrast, if we've got an observer who is stationary, so we've got, he's not moving, he stays in the same place.

02:40 He will see the clock get further away from him.

02:43 And therefore, he will see both events occurring in different positions, okay? ocar in different positions.

02:56 So events will add.

02:58 Events occur in different positions, hence not proper time.

03:07 He will see something called dilated time.

03:10 And the dilated time, but the time and observer measures between two events is given by t is equal to gamma times t0, where t0 is the proper time, and gamma is the lorentz factor, and that's equal to 1 divided by the square root of 1 minus v squared over c squared, and v is the relative velocity between the observer and the events.

03:59 So you can see here that when the observer is moving at the same at the same velocity as the clock, there is no relative velocity between him and the therefore, v in this case, will be equal to zero.

04:17 So that means gamma will be equal to 1...

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