Doppler like effect affecting observed time for a moving object

[Eismcsquared]

This is only something I’ve noticed, and in my eyes it’s odd, it may be utterly wrong or already well known but with some fancy term that I don’t know.
Suppose an object- say, a clock- is moving towards you, at about .5c. For the moment, we will assume time dilation is negligible, and that you have magical superpowers that allow you to survive the ensuing collision. (Which, at .5c, it is in this scenario.)
I’m going to attempt a diagram of sorts here.
Every “—“ will represent a distance of .2 light seconds
A “--“ represents a distance of .1 light seconds
A “-“ represents a distance of .05 light seconds
The “O” is the observer
The “X” is where the observer, well, observes the object’s location to be in the given instance of time.
The “R” is the objects real location, because in the time light reaches the observer, the object has traversed some distance in that time.
We will look at instances of time, and where these all lay in our diagram.
The object will begin at 2 light seconds, at t=0. We will say that it is already traveling at .5c, but only now begins to emit light.
t=0s
O——————————R
After 2 seconds, the light from the object, where it was at t=0, travels 2 light-seconds, and reaches the Observer. In that time, the object it self moves 1 light-second, as it is traveling at .5c.
5
t=2s
O—————R—————X
As far as the observer knows, the object turned on at this instance, and is traveling at .5c towards it. As this is a clock, we will say it measures milliseconds, and so it says “0.000” at t=2, at least as far as the observer, well, observed. However, in actuality, 2 seconds have passed already, and the clock says “2.000” at t=2, from its perspective.
Because the clock is emitting light, and at t=2, it is exactly one light-second away, then one second later the observer must observe the clock 1 light-second away. So, at t=3, our diagram looks like this:
|-.5 ls - | -.5 —l
O——--R--——X

Let’s put the next to each other
t=0 O——————————R
|————2 ls—————|

t=2 O—————R—————X
|——1 ls——|——1 ls——|

t=3 O——--R--——X—————
|-.5 ls-|—.5ls-|

With any luck, this makes sense.
As we know, v = Δd/Δt
So- at t=2 the clock appears 2ls away.
At t=3, it appears 1ls away.
v = (2ls - 1ls) / (3s - 2s)
v = 1ls/s
Last I checked, if you travel one light second in a second, you’re traveling at the speed of light!
Note- the object didn’t travel that distance, it’s just the observer that observed the clock moving that distance...

Now, remember we’re using a clock.
At t=2, the clock suddenly started showing at time of 0ms.
At t=3, it was still showing it time index, 2000ms now.
Wait a second, over 1 second, it showed a difference in time of 2 seconds! At least, to the observer...

Let’s do something different now, to see what happens in a slightly different situation. Let’s say everything’s the same, except the clock is moving away from the observer at .5c, but and starts at 1 ls.
At t=0, the clock begins displaying the time index, since when it begins to display, so 0ms.
t = 0 O—————R
1s later, the observer will see the clocks initial time index of 0 come up. By that point, the clock will have moved .5ls, as it is traveling at .5c
t=1s O—————X——--R
1.5s later, the observer will see the clock at its new locale, where it was at t=0. By then, the clock will be 2.25 ls from the observer (1.5 ls + 1.5s*.5ls/s)
t=2.5s
O———————--X———-- -R
All shown together
t=0 O—————R
t=1 O—————X——--R
t=2.5
O———————--X———-- -R
At t=1, the clock is observed 1 ls away
At t=2.5, the clock is observed 1.5 ls away
v = Δd/Δt = .5ls/1.5s = 1/3 c
Note, the clock “observed itself” at .5c, yet the observer observed the clock traveling at only 1/3 c.

Spoiler: Note!

In actuality, the clock can only observe itself as stationary, what I’m saying is that in “reality” or an unaffected time-scheme, in which light truly traveled infinitely fast and there was not time dilation, that that is what the observer would observe the clock moving at

At t=1, the observer sees the clock say 0ms
At t=2.5, the observer sees the clock say 1 second
Over 1.5 seconds, the clock appears to only change in time by one second!

Now, this seems to be entirely unrelated to time dilation. With time dilation, at .5c, the should only be a shift by a factor of only about 1.15, not 2 different factors based on relative direction, .5 and 1.5 accordingly...
This affect seems quite akin to the Doppler affect.

The equation I’ve found to represent this is as follows:
1/ (1- v/c)

Note that because v is not squared, the sign matters, where a +v is an approaching object, and a -v is and object that is growing more distant..

Another note: While this seems to break a certain law of physics, I.e. no object can travel at or faster than the speed of light, it’s probably okay here. The observer simply “thinks” the object is traveling faster than it really is. In reality, the object is traveling at whatever speed it’s traveling (v<c)

Another another note: I’d like to point out how similar this equation looks to the actual time dilation equation. Just something I find interesting.

Another another another note: Last one! I think. This is entirely theoretical, and I have no clue whatsoever as to whether it is at all correct, nor whether someone else has came upon the same conclusions I have here.
Actual last note: for real this time. Please, any suggestions are welcome! I’m simply someone in his house who asks questions and makes pathetic attempts at math, not an actually physicist! Anyone with ideas probably have better ones than mine (despite how much I’d wish otherwise :) )
And last but not least *sigh*: have a wonderful day!

 

[Nugatory]

Google for "relativistic Doppler effect" - you're right that it's interesting stuff.
Also take a look at the section of the twin paradox FAQ that describes what the earthbound twin actually physically sees.
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