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# Observing Objects near the Speed of Light

I have been thinking a lot lately about how someone would perceive objects traveling near the speed of light, and I think I stumbled upon something that no one else has realized yet. Specifically, how would an object, such as a spaceship or alien spacecraft, appear to a stationary observer if the object was traveling directly towards the observer at a velocity that is almost the speed of light?

For the sake of this article, the speed of light will be represented as "c". So, 0.5c would be half of the speed of light, and 0.75c would be three quarters of the speed of light. Also, the observer of this object is stationary on the Earth, because I believe the speed of Earths orbit and spin is insignificant when compared to the speed of light.

Now, lets imagine that an object begins its journey exactly 1 light year away from Earth. This object is heading straight for the stationary planet Earth, and it is traveling at a constant velocity of exactly 0.5c for the entire trip. Therefore, this object will arrive at Earth in exactly 2 years after it begun the journey. Now, here is where it gets a bit confusing with a lot of math. After the object travels for exactly 1 year starting from 1 light away, an observer on Earth first notices the object. The object will appear to be 1 light year away; but in reality, the object will be 0.5 light years away. This is because it took 1 year for the objects image (actually the light from the object) to reach Earth; and in that year, the object traveled 0.5 light years at a speed of 0.5c. Thus, the object has 0.5 light years remaining to reach Earth. The observer could use the Doppler Effect to calculate the actual velocity of the object and be able to calculate the objects current location assuming the observer assumed that the object did not change its course or speed. Yet, the object would still visually appear to be 1 light year away. I hope you are still with me so far.

At this point in time, the object is now 0.5 light years away, so lets imagine where the object will be in an additional 0.5 years. First, the object would have been traveling for a total of 1.5 years and have completed 75% (1.5 years divided by 2 years) of its trip. Therefore, it would have traveled exactly 0.75 light years and have a remaining 0.25 light years to complete the trip. At this time, the object would appear to the observer to be exactly 0.5 light years away, since it would have taken light exactly 0.5 years to reach the Earth when the object was 0.5 light years away.

Are you with me so far? Good, because here is the confusing part. In the 0.5 years that the observer has been observing the object, the object appeared to have traveled 0.5 light years (1 light year minus 0.5 light years). So, despite that the observer knows that the object was constantly traveling at 0.5c, the object appeared to be traveling at exactly c (0.5 light years divided by 0.5 years) from the naked eye. Of course, since nothing can accelerate up to the speed of the light; the observer may be confused with the visual observations.

Now, lets assume that the object travels an additional 0.25 years. So the object has been traveling for 1.75 years and has completed 87.5% (1.75 years divided by 2 years) of its trip. Therefore, it would have traveled 87.5 light years and have a remaining 12.5 light years to complete the trip. At this time, the object would appear to the observer to be exactly 0.25 light years away, since it would have taken light exactly 0.25 years to reach the Earth. Therefore, the object appeared to the observer to have traveled 0.25 light years (0.5 light years minus 0.25 light years) in exactly 0.25 years. Again, this is exactly the speed of light or c (0.25 light years divided by 0.25 years) that no object can accelerate to.

If we change the initial velocity of the object to 0.75c instead, the observer will observe that the object is traveling faster than at the speed of light. For example, after the first year of travel, the object would have traveled 0.75 light years and be 0.25 years from Earth. The observer would see the object to be 1 light year away. In the next 0.25 years, the object would have traveled an additional 0.1875 of a light year (0.25 years times 0.75c), so the object would have traveled a total distance of 0.9375 light years (0.75 plus 0.1875 light years) in a total of 1.25 years. The observer would observe that the object to be 0.25 light years away. Thus, the observed distance that the object traveled in the last 0.25 years would have traveled 0.75 light years. Therefore, the object is perceived to be traveling at an amazing 3 times the speed of light (3c = 0.75 light years divided by 0.25 years) to the observer!

Another way of explaining this phenomenon is the object is approaching the observer at such as large percentage of the speed of light that for each time the observer visually sees the object, it takes significantly less time for the objects light to reach the observer. Therefore, the observer perceives that the object is traveling at a faster velocity than what it is actually traveling. Thus, it is possible to perceive an object to be traveling faster than the speed of light when it is not.

If you think my theory is incorrect, then please email me your reason why my logic or math is incorrect. Thank you!

by Phil for Humanity
on 07/11/2007