# Time Dilation with objects at low speeds

Emc2fma
First post on PF!

Anyways, I'm a high school student so please forgive my lack of knowledge in some areas :)

So I was reading about special relativity and I understand the ideas behind both postulates. It's the application of those postulates that confuses me.

For example, please let me know if my idea of time dilation is correct. So if a person is sitting on a train with a flashlight and turns it on, the speed of that beam of light will be 186,282 mps. The additional movement of the train is not added to the speed of light because nothing (with mass) can travel faster than light (because of infinite energy). So since the speed of light cannot change, light accounts for for the added velocity of the train by slowing down time itself.

Is that correct? Because the way I think of it, a stationary flashlight and a moving flashlight (both are turned on at the same position) will give off a beam of light towards an object that has the same speed.

Since speed = distance / time, and both speed and distance are the same for both flashlights, time is the only variable that can change for the moving flashlight. Is this correct? But then this doesn't explain why the stationary flashlight also experiences time dilation.

If I have this part correct, then I'll ask the REAL question I had (which is related to low speeds as mentioned in the title).

Homework Helper
First post on PF!
Welcome to PF!
So I was reading about special relativity and I understand the ideas behind both postulates.
Well done - it can take some people many years ;)
It's the application of those postulates that confuses me.

For example, please let me know if my idea of time dilation is correct. So if a person is sitting on a train with a flashlight and turns it on, the speed of that beam of light will be 186,282 mps. The additional movement of the train is not added to the speed of light because nothing (with mass) can travel faster than light (because of infinite energy).
Close - the infinite energy thing, in the derivation, is a consequence of the second postulate - not the reason for it. This step goes: if the second postulate is true then the motion of the train does not affect the speed of light.

Fire a light beam through a train - the people on the train will agree with the people on the tracks about what the speed of the light in that beam was (but they'll disagree about it's color.)

So since the speed of light cannot change, light accounts for for the added velocity of the train by slowing down time itself.
Nope. People on the train are not in some weird "slow time" world. As far as they are concerned, it is the people on the track that have the slow time.
It is not the presence of the light that creates time dilation - that would happen in the dark just as well.

Is that correct? Because the way I think of it, a stationary flashlight and a moving flashlight (both are turned on at the same position) will give off a beam of light towards an object that has the same speed.

Since speed = distance / time, and both speed and distance are the same for both flashlights, time is the only variable that can change for the moving flashlight. Is this correct? But then this doesn't explain why the stationary flashlight also experiences time dilation.
There is no such thing as "the stationary flashlight" ... the observer never experiences time dilation ... it is always something that is happening to other people. If the flashlight is moving, with respect to the observer, then it's clock ticks slower than the observer's clock, from the POV of that observer.

If two observer's are moving with respect to each other, then they will each see the other's watch moving slowly ... this has to be the case or the second postulate is wrong.

It's hard to wrap you head around I know.
There is a neat introduction that may help.

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Mentz114
Welcome.
You are correct in that the speed measured for light by any observer will be independent of the relative velocity of the source of the light. I'm not sure about the rest. Usually the dual effects of time dilation and length contraction are cited. I recommend you look at the Lorentz transformation which changes coordinates between two observer.

[Posted at the same time as the above]

Naty1
Hi EMC

So I was reading about special relativity and I understand the ideas behind both postulates.

Don't be so sure!
What I mean is that those basics will have new meaning to you, I bet, after you finish a few discussions in these forums. Those basics 'sound easy' at first but I sure had a lot to learn after I thought I understood them. But don't feel bad, it took an 'Einstein' to figure them out and develop a theory. And he could not even solve his own equations! [That's why exact solutions in general relativity have names like Schwarzschild, Reissner–Nordström, etc, to commemorate those who did solve them!]

For example, please let me know if my idea of time dilation is correct. So if a person is sitting on a train with a flashlight and turns it on, the speed of that beam of light will be 186,282 mps. The additional movement of the train is not added to the speed of light because nothing (with mass) can travel faster than light (because of infinite energy). So since the speed of light cannot change, light accounts for for the added velocity of the train by slowing down time itself.

Is that correct?

no...you realize the flashlight 'beam' has no rest mass, right?? Light does not 'account' for a change in time.

And easy rule to remember is: everybody observes the local speed of light as 'c'. So you cannot 'add to the speed of light'...

I'll bet you are thinking that two velocities [u and v] normally add as in w = u + v.

That is an ONLY approximation and works well for speeds way below c, like trains and planes.
The actual formula is here:

To see what happens, in the correct formula, let one of the speeds [u or v] ] equal c [say u =c]. This would be 'adding lightspeed c to v'. Then simplify the right hand side...what do you get: c! You cannot go faster than lightspeed 'c'.

Regarding time: it turns out that while the speed of light in special relativity IS fixed for all nearby [coincident] inertial observers, a constant time and distance which SEEM to us in daily life like they are constant, are NOT...they are dynamic variables! Since they vary, you cannot directly and precisely add u and v together because time for each elapses differently and distance does too. [Changing time is called time dilation and changing length is called length contraction.]

Good luck these forums can be fun!

Emc2fma
Thank you to everyone who responded!

@Simon, That link you posted really sealed the deal and helped me understand my second question as well! Thanks!

I think special relativity is one of the most interesting things I have EVER learned. Both you and Naty said that even though I "think" I understand special relativity, I "really" don't.

What other topics (besides the time dilation/length contraction) should I read about in your opinion(s)? Haha I really want to understand special relativity :)

Mentor
What other topics (besides the time dilation/length contraction) should I read about in your opinion(s)? Haha I really want to understand special relativity :)

It's actually pretty easy, as long as you steer clear of the pop-sci books and videos (these tend to focus too much on the the "gee-whiz" time dilation, length contraction stuff, too little on the thinking behind them).

Here's a basic self-study sequence for a competent layman's understanding, which will put you ahead of 99.9+% of the population and qualify you to answer a fair number of the simple relativity misunderstandings that show up in this forum.

The only prerequisite is high-school algebra (without that minimal math background, you'll find yourself having to take somebody else's word for EVERYTHING in science).

1) Start with Einstein's two postulates - you've done that. For extra credit, read some of the history around them, especially the problems that the speed of light created for physicists during the second half of the nineteenth century.

2) Learn how to derive the Lorentz transformations from these two postulates. For present purposes (I'd give you a different recommendation if you were working on a career in physics) it's hard to improve on the derivation in the appendix of Einstein's book "Relativity - The Special and General Theory".

3) Learn how to derive the relativity of simultaneity by using the Lorentz transforms to relate what the two observers in the train-and-lightning thought experiment see. Draw a basic space time diagram with the coordinate axes of the stationary and the moving observer marked on them - and then draw another one the from the perspective in which the train observer is at rest and the platform observer is moving in the opposite direction.

4) Use the Lorentz transforms to derive time dilation and length contraction. For extra credit, derive the relativistic rule for adding velocities, but it's OK to just accept that formula as a given.

5) Now check your understanding by trying the pole-barn "paradox". If you think you could competently explain it to someone else, you're there.

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Emc2fma
^Thank you for those suggestions! I'll take a look at those as soon as I figure out the answer to something that's been bothering me all day today (and try as I might, I can't understand it).

I'm trying to understand why relativistic mass increases as velocity increases. Just to clarify, rest mass is the "innate" property of an object relating to its matter (ie. doesn't change) while relativistic mass depends on the perception of that object. Is that right?

I've heard many people say that as you increase speed, it requires more and more energy because the relativistic mass also increases. Relativistic mass is infinite when an object with rest-mass is traveling at the speed of light. So to move an object with infinite mass, you'll need infinite energy (which isn't possible) so you can never go past the speed of light.

WHY?

I've seen the equation which demonstrates why mass reaches infinity when velocity is c, but I simply don't understand it conceptually. Can anyone explain? Why the heck does your perception of an object's mass increase at faster speeds? (Unless I have a completely wrong definition of relativistic mass)

Gold Member
Emc2fma

The total energy of an object of rest mass m and velocity v relative to an observer is given by$$E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}$$For small values of v this approximates to $E \approx mc^2 + \tfrac{1}{2}mv^2$. For large values of v it approaches infinity as v approaches c.

I hope you are already aware of the concept of kinetic energy in Newtonian physics. The idea of relativistic mass is to use E=mc2 and treat kinetic energy as mass, i.e. the increase in mass is just the increase in kinetic energy, expressed as mass instead of energy.

The concept of relativistic mass is out of fashion amongst professional physicists who prefer to work with rest mass only (which they call just "mass").

As to why the above formula for energy is true, you'll need to study some books.

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Mentor
I've seen the equation which demonstrates why mass reaches infinity when velocity is c, but I simply don't understand it conceptually. Can anyone explain? Why the heck does your perception of an object's mass increase at faster speeds? (Unless I have a completely wrong definition of relativistic mass)

Several answers, in no particular order:
1) Strictly speaking, mass does not reach infinity when velocity reaches c, because velocity never reaches c. If you were to substitute the words "when pigs fly" for "when velocity reaches c", in the italicized text above, it wouldn't change the meaning. (See also #3 below).

2) You are tripping over another of the confusions that happen when you start with the gee-whiz stuff (length contracts, time dilates, mass increases) instead of starting with the Lorentz transformations from which these effects can be predicted.

3) It's sloppy to say that "mass increases with velocity". It is much more accurate to say that when you measure the mass of a object that is moving relative to you, you will get a result that is greater than the mass measured by an observer at rest relative to the object. That is, nothing odd happens to the object - it's just that different observers measure different masses for the object, just as they measure different lengths and different clock tick rates. This takes us to #4...

4) How do we measure the mass of an object? We apply a force F to it, measure the resulting acceleration a, plug these two values into the equation F=ma, and out pops the mass m. (Think about how a scale works if you don't believe me). Now consider me, watching an object moving away from me under the influence of a constant force... No matter how long that force acts, the object will never exceed the speed of light so the acceleration we measure for it will go down as it gets nearer to the speed of light. If the acceleration is going down, the force is staying the same, and F=ma holds... Then our measurement is showing an ever-increasing value of the mass.

5) The entire concept of relativistic mass, as described in #4 above, has fallen out of favor in modern physics. There's nothing inherently wrong or untrue about it, but as a mathematical description it's useful only for problems that involve applying a force to an object that is moving rapidly relative to the source of the force (relativistic electrons moving through an electrical field generated by lab equipment would be an example). Thus the modern style is to use the more general equation $$E^2=(pc)^2+(mc^2)^2$$ where E is the energy, m is the rest mass, and p is the momentum. This equation encapsulates both the old relativistic mass equation (remembering that p is defined as a response to an applied force) and the famous $E=mc^2$