Do you have to travel away from the observer for time to slow down?

[zeromodz]

From my intuitive perspective on time dilation. The faster an observer is going relative to another, the slower in time the observer moves with respect to the to the other. However, I always pictured that this only works if the observer is moving away with respect to an observer at rest. Since velocity is a vector, would the direction matter in lorentz transformations? Because Gamma contains v/c, not speed, but velocity.

 

[Fewmet]

From my intuitive perspective on time dilation. The faster an observer is going relative to another, the slower in time the observer moves with respect to the to the other. However, I always pictured that this only works if the observer is moving away with respect to an observer at rest. Since velocity is a vector, would the direction matter in lorentz transformations? Because Gamma contains v/c, not speed, but velocity.

You square the velocity in gamma, so you get the same result with v and -v. If it helps, it is like squaring the velocity in calculating kinetic energy.

 

[Doc Al]

However, I always pictured that this only works if the observer is moving away with respect to an observer at rest.

No, that's not correct.

Since velocity is a vector, would the direction matter in lorentz transformations?

Not for time dilation. But length only contracts in the direction of motion.

Because Gamma contains v/c, not speed, but velocity.

True, but gamma contains v/c squared, which you can think of as a scalar product. So the direction of motion does not affect gamma; only the speed does.

Direction does matter in the general Lorentz transformations, but not because of gamma. (The convention is to take the primed frame to be moving along the +x axis.)

 

[George Jones]

Maybe the following is relevant.

In reality, the phrase "a moving clock runs slow" does not necessarily mean "a moving clock is seen visually to run slow." A clock moving directly away from an observer appears visually to run slow, but a clock moving directly towards an observer appears visually to run fast. In both cases, what is seen visually is given by the Doppler expression, which is always different than the time dilation expression. In both cases, the time dilation expression, used appropriately, does apply.

Consider the following example.

Assume that Alice is moving with constant speed directly towards Ted. When Ted uses his telescope to watch Alice's wristwatch, he sees her watch running at a faster rate than his watch. Ted sees Alice's moving watch running fast, not slow! Ted sees this because of the Doppler shift. Because Alice moves towards Ted, the light that Ted sees from her watch is Doppler-shifted to a higher frequency. But the rate at which a clock or watch runs is like frequency, i.e., a second-hand revolves at a certain frequency, and all frequencies are Doppler-Shifted., so ted see Alice's wristwatch running fast.

To explain what "A moving clock runs slow." means, I first have to explain how Ted (with help from Bob) establishes his frame of reference.

Starting from Ted, a series of metre sticks, all at rest with respect to Ted, are laid end-to-end by Bob along the straight line joining Alice and Ted. At each joint between two consecutive metre sticks, Bob places a small clock. The metre sticks and clocks all are at rest with respect to Ted. Initially, none of the clocks are running; before turning them on, the clocks have to be synchronized. To do this, Ted directs a laser pointer along the line joining Ted and Alice, and then sends a flash of light. Each clock is turned on when the flash of light reaches it. The speed of light is not infinite, so the time taken for the light to travel from Ted to each clock has to be taken into account. To do this, the clocks' hands are set initially as follows. The clock one metre away from Ted is set to the time taken for light to travel one metre; the clock two metres away from the tower is set to the time taken for light to travel two metres; ... .

This whole setup of metre sticks and clocks establishes Ted's reference frame.

Now, As Alice moves toward Ted, Ted uses his telescope to watch Alice's wristwatch, and to watch his clocks. First, he watches one of the distant clocks in his reference frame. The time he sees on the clock is the time at which the light he sees set out from the clock, so Ted sees an earlier time on the distant clock than he sees on his wristwatch. Because the clock is stationary in his frame, Ted does, however, see the distant clock running at the same rate as his watch. Similarly, Ted's sees all the clocks in his frame running at the same rate as his watch.

As Alice approaches Ted, she whizzes by clock after clock of Ted's reference frame. Using his telescope, Ted sees that Alice is beside a particular clock, and he notes the time on her watch and the time on the clock adjacent to her. Some time later, Ted sees Alice beside a different clock, and he again notes the time on her watch and the time on the clock adjacent to her.

Ted checks his notes, and he finds that the time that elapsed on Alice's watch as she moved between these two clocks of his frame is less than the difference of the readings of the two clocks. This what is meant by "A moving clock runs slow."

Unfortunately, "time dilation" in general relativity and "time dilation" in special relativity often have different operational meanings. Suppose observer A hovers at a large distance from a Schwarzschild black hole, and that observer B hovers near the event horizon. If observer A uses a telescope to observe B's watch, A will see B's watch running more slowly than his own watch. In this context, gravitational time dilation is something that is seen visually.

 

[Passionflower]

From my intuitive perspective on time dilation. The faster an observer is going relative to another, the slower in time the observer moves with respect to the to the other. However, I always pictured that this only works if the observer is moving away with respect to an observer at rest. Since velocity is a vector, would the direction matter in lorentz transformations? Because Gamma contains v/c, not speed, but velocity.

No, moving away and approaching gives the same time dilation effect.

Also, relativity works in mysterious ways, a rocket accelerating away from a planet may under some circumstances get actually closer and closer to this planet.

 

[GrayGhost]

From my intuitive perspective on time dilation. The faster an observer is going relative to another, the slower in time the observer moves with respect to the other. However, I always pictured that this only works if the observer is moving away with respect to an observer at rest. Since velocity is a vector, would the direction matter in lorentz transformations? Because Gamma contains v/c, not speed, but velocity.

zeromodz,

The relative rate of time is defined by the gamma factor ...

Tau = gamma*t

gamma = 1/(1-v²/c²)^1/2​

No matter what the direction of v, the gamma factor remains the same. It does not depend upon the direction of relative motion, so really it depends only on the magnitude of the velocity, ie the speed. Also, since v is squared ... if one were to consider the motion oppositely (-v instead of v), it's doesn't matter since v² = (-v)² ... and so the gamma factor does not change.

GrayGhost