# Hypothetical thought experiment about time dilation

1. Apr 7, 2013

### awalltep81

Let's say that you leave Earth at 75% the speed of light (c). You travel out and back for a minute. And when you come back an hour has passed on Earth. (I understand this is not accurate; I'm just using it for the sake of this hypothetical) You leave again at 99% of c, again for a minute. When you get back to Earth, a day has passed. You do it again at 99.99999% of c. You come back to Earth and 10 years has passed. Again at 99.99999999999% and 10,000 years has passed. You can keep adding 9 onto the end of that percentage, and the one minute you travel will be more and more time for those left behind.

Isn't it true, then, if you are actually able to hit the speed of light, that means upon "returning" you've completely bypassed time itself? Shouldn't an infinite amount of time have passed? Since you can "bypass" (can't think of a better word) any measurable amount of time by getting closer and closer to c, wouldn't it make sense that to actually hit c would be to "bypass" an immeasurable amount of time?

2. Apr 7, 2013

### 1977ub

It turns out you will not be able to accelerate to the speed of light relative to the Earth. The more energy you expend from your engine, the faster you will get, up toward - but not including - the speed of light.

3. Apr 7, 2013

### awalltep81

I know you can not actually reach the speed of light. It's kind of a "what if"?

4. Apr 7, 2013

### HallsofIvy

Yes, relativity says it is impossible for any "massive" object to move at the speed of light. So even in a "thought experiment" you cannot say "if you are actually able to hit the speed of light". You are essentially asking "if relativity were wrong, what would relativity say about this situation?"

5. Apr 7, 2013

### 1977ub

The faster you go, the slower your clock ticks relative to Earth. For "an infinite amount of time" to pass, you would have to travel while ∞ time passed on Earth. Presumably the Earth will not last that long. There might be a big crunch.

[edited to turn it around]
The larger and larger you make x, the smaller and smaller 1/x becomes. Doesn't it make sense that for x to "actually hit" infinity, then 1/x would be zero? Anyhow, no mass can ever be accelerated to the speed to light.

Last edited: Apr 7, 2013
6. Apr 7, 2013

### awalltep81

Ok. I never realize that flaw in the hypo. Thanks.

7. Apr 8, 2013

### ghwellsjr

Just for the fun of it, I'm going to modify your question to bring it in line with reality. I realize you were not concerned with the details so I'm not faulting you on this point.

If we start with a speed of 99% of c, the time that will pass on Earth will be 7.088812 minutes. If, instead of adding a single 9 onto the end of the percentage, we add a pair of nines, the time will go up by about a factor of 10. So at 99.99%c, 70.712446 minutes will transpire on Earth. At 99.9999%c, it is 707.106958 minutes on Earth. We can see a pattern here and if we realize that the factor approaches the square root of one-half (0.70710678118654752440084436210485) times a power of ten, then we can easily predict what the factor is for speeds as close to the speed of light that we desire. But no matter how many pairs of nines we want to add onto the end of the percentage, the digits in the factor will always approach that of the square root of one-half. So, for example, with 30 pairs of nines in the percentage of the speed of light, the factor will be 707106781186547524400844362104.85 minutes. Now that is a very long time, much longer than the age of the universe but still no where near infinity.