Time dilation - Is the time between events longer or shorter

[bookerdewitt]

So let's say there's a train and according to the passengers on the train the trip takes a time t to go from the station and back. According to an observer at the station, would they measure a time interval of gamma *t or t/gamma?

I know they both should see the others clock ticking at a slower rate, so does that mean each observer's own clock will be ticking at a faster rate relative to the other one, which would mean the time difference as measured by the observer at the platform would be longer?

 

[andrewkirk]

would they measure a time interval of gamma *t or t/gamma?

It will be neither, because the train does not have a constant reference frame, because it goes away and back again -- in which case it must have been accelerated at the turnaround point, as well as at the start and beginning of the journey. Hence this situation is too complex to be represented using just the time dilation factor. It is in fact a version of the twin paradox.

If you want to use time dilation you need the train to be in constant, unaccelerated motion.

 

[bookerdewitt]

Ok sorry, suppose the train just travels there and is moving at constant velocity. Would the trip duration be longer for the observer at the platform?

 

[FactChecker]

When you add "and back" you bring the issue of deceleration and accelerating to return. That complicates things. The Twins Paradox is based on the confusion.

If the train is going in one direction and not changing speed, the situation is different. Each person, one on the train and one on the ground, thinks that the other persons clock is going slower than it should. ("going slower" means that each has set up his own set of clocks along the length that he thinks are synchronized and the other person observes those clock times as they pass by as though they were one clock ticking)

 

[andrewkirk]

Say both the platform observer P and the train observer P' start their stopwatches when the train's front passes the end of the platform at station S1 and stop them when the the train's front passes the end of the platform at station S2. If T and T' are the times on their two stopwatches then T' = T/gamma. So T'<T.

The apparent conflict with the fact that P' observes the clock of P ticking more slowly than their own clock is resolved by the fact that in the frame of P', P doesn't stop her watch until long after P' stops hers.

 

[bookerdewitt]

Ok, so the time that has passed between two events as measured by the stationary observer will always be larger

Say both the platform observer P and the train observer P' start their stopwatches when the train's front passes the end of the platform at station S1 and stop them when the the train's front passes the end of the platform at station S2. If T and T' are the times on their two stopwatches then T' = T/gamma. So T'<T.

The apparent conflict with the fact that P' observes the clock of P ticking more slowly than their own clock is resolved by the fact that in the frame of P', P doesn't stop her watch until long after P' stops hers.

 

[andrewkirk]

Ok, so the time that has passed between two events as measured by the stationary observer will always be large

No. ONly between those two events. That generalisation is not valid. The addition I made to my last post, via edit, may help clarify that.

 

[bookerdewitt]

So how do you know whether to divide or multiply by gamma? I thought that if you pick the stationary observers frame to measure the time in, you always multiply the time measured by the moving observer by gamma.

 

[andrewkirk]

The rule is that, if P measures the time T in the frame of P and observes the time ticked off by the clock of P', between spacetime events E1 and E2, then T' = T / gamma, so T' < T.

We have three key spacetime events. E1 is both observers starting their watches, E2 is P' stopping her watch and E3 is P stopping her watch. In the frame of P, E2 and E3 are simultaneous. But in the frame of P', E2 occurs before E3. In analysing the problem you need to make clear whether you are measuring the time from E1 to E2, or from E1 to E3. If that is not made clear, as is the case in your rendition of the problem, confusion arises.

 

[bookerdewitt]

Ok I think I understand now. Thanks.

 

[15characters]

Ok I think I understand now. Thanks.

I think you are confused because from the train's perspective the following statements are true

1) the station clocks are slow, and
2) the station clocks record more time for the journey

Clearly, you are thinking if a clock system is slow how can it record more time? It seems like a paradox. Its not a paradox. Put simply...

From the station's perspective

• Train clocks are slow

• Station clocks are all synchronised

From the train's perspective

• Station clocks are slow

• Station clocks are NOT synchronised.

As I see it you are fine with the clock's moving slow. Now the reason for your confusion "Why do the slow station clocks record more time?".

It's because from the train's perspective, the station clock's are NOT synchronised.https://en.wikipedia.org/wiki/Spacetime
https://en.wikipedia.org/wiki/Proper_time
https://en.wikibooks.org/wiki/Special_Relativity/Simultaneity,_time_dilation_and_length_contraction

 

[15characters]

Most useful bits...copy pasted

Events

The basic elements of spacetime are events.
, an event is a unique position at a unique time.

Coordinate systems

in describing physical phenomena (which occur at certain moments of time in a given region of space), each observer chooses a convenient metrical coordinate system.

Spacetime interval

In a Euclidean space, the separation between two points is measured by the distance Δr between the two points. Whereas...

In spacetime, the separation between two events is given by the distance Δr and the time difference Δt

The spacetime interval, is s2, is :

(r = distance and t = time)

S2 is also called the invariant interval because it does not change for any observer in any frame of reference. r and t may change but s will remain the same. Time-like interval

For two events separated by a time-like interval, enough time passes between them that there could be a cause–effect relationship between the two events. For a particle traveling through space at less than the speed of light, any two events which occur to or by the particle must be separated by a time-like interval. Event pairs with time-like separation define a negative spacetime interval ([PLAIN]https://upload.wikimedia.org/math/6/1/c/61c019b207c743577d85d4a65f5a583c.png) and may be said to occur in each other's future or past. There exists a reference frame such that the two events are observed to occur in the same spatial location, but there is no reference frame in which the two events can occur at the same time.

The measure of a time-like spacetime interval is described by the proper time interval, [PLAIN]https://upload.wikimedia.org/math/3/5/5/355deec8daaddf14b3d7c610cb90e75a.png:

(proper time interval).
The proper time interval would be measured by an observer with a clock traveling between the two events in an inertial reference frame, when the observer's path intersects each event as that event occurs. (The proper time interval defines a real number, since the interior of the square root is positive.)

Light-like interval
In a light-like interval, the spatial distance between two events is exactly balanced by the time between the two events. The events define a spacetime interval of zero ([PLAIN]https://upload.wikimedia.org/math/6/6/0/6602be59bc8c0b4a73716ac3e62afa31.png). Light-like intervals are also known as "null" intervals.

Events which occur to or are initiated by a photon along its path (i.e., while traveling at [PLAIN]https://upload.wikimedia.org/math/4/a/8/4a8a08f09d37b73795649038408b5f33.png, the speed of light) all have light-like separation. Given one event, all those events which follow at light-like intervals define the propagation of a light cone, and all the events which preceded from a light-like interval define a second (graphically inverted, which is to say "pastward") light cone.

Space-like interval

When a space-like interval separates two events, not enough time passes between their occurrences for there to exist a causal relationship crossing the spatial distance between the two events at the speed of light or slower. Generally, the events are considered not to occur in each other's future or past. There exists a reference frame such that the two events are observed to occur at the same time, but there is no reference frame in which the two events can occur in the same spatial location.

For these space-like event pairs with a positive spacetime interval ([PLAIN]https://upload.wikimedia.org/math/6/3/0/630969e26cd6cbfdbda5ffe131096b10.png), the measurement of space-like separation is the proper distance, [PLAIN]https://upload.wikimedia.org/math/e/7/d/e7d12a405a591e43b2bd20cc74de2411.png:

(proper distance).

Like the proper time of time-like intervals, the proper distance of space-like spacetime intervals is a real number value.

 

[bookerdewitt]

Thanks a lot! This definitely helps clear up some misunderstandings I had.