Can Classical Mechanics Calculate Time Dilation?

[AndromedaRXJ]

Okay so I have a question about time dilation, kinetic energy and classical mechanics. My question is, if an object were traveling at very high relativistic speeds and experienced time dilation, would the time it experience and measure during the travels be equivalent to the travel time if it were traveling in a classically mechanical universe where FTL is possible? Assuming the same amount of energy is used the accelerate the object in both cases?

For instance, say I have a small space pod that weighs 1000 kg. And classically (no relativity) If I were to accelerate the ship to, say, 5c, the KE of the ship should be 1.125e+21 joules. And if I'm five light-years from some location that I'm approaching, then my travel time will be 1 year until I arrive at that location.

So now relativistically, I have the same ship (1000 kg) and I spend 1.125e+21 joules to accelerate it. I know it won't be FTL, but my question is, if I travel 5 light years in roughly 5 years, will I only experience 1 year due to time dilation?

Basically, my question is, can I use classical mechanics to calculate classical FTL travel times to see what my time dilation would be relativistically with the same amount of KE?

 

[Vanadium 50]

I use classical mechanics to calculate classical FTL travel times to see what my time dilation would be relativistically with the same amount of KE?

No. The equations are different.

 

[Nugatory]

if I travel 5 light years in roughly 5 years, will I only experience 1 year due to time dilation?

Only for one particular speed.

Say you are traveling to a star at rest relative to the Earth and five light-years away according to an observer at rest relative to the Earth and the star. Whether you travel at .999c, .999999c, or .999999999c you will cover the distance in roughly five years according to that observer... but the elapsed time that you measure will be very different in the three cases.

 

[AndromedaRXJ]

No. The equations are different.

I know this. The kinetic energy is already given in both situations though. Let my try to ask the question differently since I think I asked it poorly.

Say a particle in a classical universe crosses 5 light years in 1 year, there for it also observes it self as traveling for 1 year. It's kinetic energy and mass are known.

Now say in a universe where relativity applies. The particle has the same mass and kinetic energy. Is the time it measures for it self 1 year? I know that's not what stationary observers will observe, but is that what it measures for it self?

 

[AndromedaRXJ]

Only for one particular speed.

Say you are traveling to a star at rest relative to the Earth and five light-years away according to an observer at rest relative to the Earth and the star. Whether you travel at .999c, .999999c, or .999999999c you will cover the distance in roughly five years according to that observer... but the elapsed time that you measure will be very different in the three cases.

I know this. The mass and kinetic energy are already given, so the speed is already known. I'm wondering if the travel time it measures for it self is the same in both situations. I'm not concerned with stationary observers.

 

[Vanadium 50]

The particle has the same mass and kinetic energy. Is the time it measures for it self 1 year?

The answer is still no, for the same reason.

 

[AndromedaRXJ]

The answer is still no, for the same reason.

Is the time it observes for it self less or more than what it would observe classically? Or does that vary?

 

[PeterDonis]

if I travel 5 light years in roughly 5 years, will I only experience 1 year due to time dilation?

No; you will experience significantly less than 1 year. The math is simple for this, but it's not the simple math you appear to be hoping for.

You have a rest mass of 1000 kg, which is a rest energy of $9 \times 10^{19}$ Joules (rest mass times $c^2$, which is about $9 \times 10^{16}$). You have a kinetic energy of $1.125 \times 10^{21}$ Joules. Your total energy is the sum of the two, which is $1.215 \times 10^{21}$ Joules. Your relativistic gamma factor, which is your time dilation factor, is the ratio of your total energy to your rest energy, which is $121.5 / 9 = 13.5$. So you will experience $5 / 13.5 = 0.37$ years of elapsed time.

 

[AndromedaRXJ]

No; you will experience significantly less than 1 year. The math is simple for this, but it's not the simple math you appear to be hoping for.

You have a rest mass of 1000 kg, which is a rest energy of $9 \times 10^{19}$ Joules (rest mass times $c^2$, which is about $9 \times 10^{16}$). You have a kinetic energy of $1.125 \times 10^{21}$ Joules. Your total energy is the sum of the two, which is $1.215 \times 10^{21}$ Joules. Your relativistic gamma factor, which is your time dilation factor, is the ratio of your total energy to your rest energy, which is $121.5 / 9 = 13.5$. So you will experience $5 / 13.5 = 0.37$ years of elapsed time.

Wow that is really helpful, thanks! But you're right though. I was hoping for observed times for classical and relativistic travelers to correspond for a given mass and kinetic energy. I thought that would have been interesting if true, and would have been a neat math trick.