MacM:
Let me see if I can work out what you're doing.
For simplicity, let's assume you accelerate your car from 0 to 108 km/hr in 10 seconds.
108 km/hr is conveniently v=30 m/s.
The gamma factor at that speed is
$$\gamma = \frac{1}{\sqrt{1-(v/c)^2}}$$
Since v is very small compared to the speed of light, we can approximate this as:
$$\gamma \approx 1 +\frac{1}{2}\left(\frac{v}{c}\right)^2$$
I think this is what you were doing with your formula.
Plugging in the numbers v=30 m/s and c = 300,000,000 m/s, we have
$$\gamma = 1.000000000000005$$
Now consider, say, a galaxy that is 100 million light years away when you're at 30 m/s, as measured in the reference frame of the car.
At a speed of 30 m/s, the distance to the galaxy is d (as measured by you, in the car), as compared to d0 when you were at rest, and
$$d = d_0 / \gamma$$
Therefore:
$$d_0 = \gamma d = (\gamma - 1)d + d$$
So
d0 = (0.000000000000005)(100,000,000) + 100,000,000 light years
d0 = 100,000,000.0000005 light years.
Thus, in 10 seconds, the distance to the galaxy has shrunk by 0.0000005 light years, or around 4.73 times 10^9 metres.
The "speed of shrinkage" is this distance divided by the time taken to shrink, which gives 4.73 times 10^8 metres per second.
But the speed of light is only 3.00 times 10^8 metres per second! So, the rate of distance shrinkage means that the shrinkage happened faster than light.
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Is this your argument?
And do you think this is a problem for relativity? If so, why?
Well James R, I'm very impressed and pleased. In all the years you and I have locked horns this is the first time you actually took time and made effort to address the actual issue rather than just slam the post.
That being said I would say being a problem for SR is debatable. Many (likely you included) will be willing to ignore the implications because SR is otherwise useful.
I however, do think it points to the falsification of the concept as physical reality. After all the velocity one gets while accelerating is now a function of the distance to the object and the energy you are expending to achieve a velocity is no longer equally proportional - i.e. BTU's/Mph etc.
That is locally I can say I consume "1,000BTU's to go from 0 - 60Mph but those same 1,000BTU's also take me from 0 - 97,699,556.54 Mph to the gallaxy 100 million lyr away.
Velocity now becomes a completely arbitrary value.
SR actually addresses this issue and uses event horizons to mask the FTL motion. My point is that MOST folks here that think they understand SR actually have never thought about some of the arcain consequences of the concept.
One such arcain consequence also being the fact that accelerating AWAY from a remote object you actually accelerate closer. Now vector of motion is lost.
This failure to critically consider the consequences of SR is systemic in modern physics and astronomy. For example how many people have ever considered that what we observe as the accelerating expansion of the universe may well be the consequence of inverse lorentz contraction of a decelerating universe (this is assuming you retain spatial contraction as a reality - I don't).
None of this occurs if you simply acknowledge that a dilated (accelerated and now having a different absolute velocity) clock is physically ticking slower than the resting inertial clock and that it's accumulated time over the course of a trip compared to the time recorded by the resting clock fully accounts for the trip with NO spatial contraction having taken place.
That you create spatial contraction by ignoring the dilated tick rate and claiming the less time is because you traveled less distance at a common velocity. Frankly it is an absurd concept and has never been observed. Declaring a physical affect in one frame and a different physical affect in another. It is not necessary and generates absurdities.
Stick with one physical affect.
If I have accelerated and have established a velocity to another resting clock each meter marker I pass will seem to require less time; hence I will compute I have a higher velocity, not that the meter marker physical distance shrank and I have a common velocity.
Velocity is a computed value based on the relationship of two physical attributes - Time and Distance. Velocity is not a physical quantity. It is incorrect to claim a common computed value when you have already determined that one physical quanity has changed.
That is my arguement in a nut shell. SR makes a wrong turn in it's conclusions and basis.
It is not distance that changes physically, it is velocity that is frame dependant because it is a computed value based on two different sets of physical facts - the change in clock tick rate only. v = d/t.
Use t1 = t2/2 and you can see that v1 = 2v2. Distance did not change.
