Does gravity have a similar effect on energy as it does with time?

[mcjosep]

I've been doing some recreational physics and wanted to see if I could come up with a formula for energy/time equivalence. I decided to start with the formula for gravitational time dilation.

$$
y=Z\sqrt{1-(2GM/rc^2)}
$$

This formula shows what time would look like to an outside observer (infinitely far) looking in at a watch within a gravitational field. $Z$ is the time of the observer and $y$ is the slow moving time in a gravitational field at radius from the mass $r$.

I modified the formula a little to look for energy change, this is probably where a mistake was made.

$$
(1-(y*5.39106*10^{-44})))=1*\sqrt(1-(2*6.67384*10^{-11}*(x/299792458^2))/(1.616199*10^{-35}*299792458^2))
$$

what you see in the above formula to the left of the equal sign (assuming we are comparing a single second for the top formula $Z$ to the dilated time) is 1 second - x*Planck time.

to the right of the equal sign everything is the same except rather than the $M$ for mass in the top formula we used $E/c^2$ from the mass energy equivalence formula $E=mc^2$ so the $x$ in the above formula represents Energy.

For radius, $r$, I fixed it at the Planck length.

So when I entered the formula above into my favorite tool WolframAlpha to solve for $x$ it returned.

$$
x=1.05457*10^-34*y-2.84263*10^-78*y^2
$$

what this is showing is

$$
E=\hbar*y-\hbar*P_t*2*y^2
$$

(Reduced Planck constant * the amount of plank time units you want to take away from one second) - (Reduced Planck constant * 2 * Planck time * the amount of plank time units you want to take away from one second ^2) equals the multiple at which energy would change.

I believe the "2" is in the formula above because at the same rate time would be effected by gravitational time dilation so would length, so you multiply the Planck time by 2 and you get the effect of both being "dilated" equaling the rate at which energy changes. This is just a guess, I know that that gravitational length contraction is a disputed subject.

I put the formula above so that you could drop it in wolfram yourself and see the answer, I do not have all the steps it took to derive. I was mainly surprised how my modified formula turned into a formula with two constants and a variable.

I was also surprised how it charted out. Let me know what you think.

[Modified formula][1]

[Derived Formula][2] [1]: http://www.wolframalpha.com/input/?...^2))/(1.616199*10^-35*299792458^2)) for x&f=1
[2]: http://www.wolframalpha.com/input/?i=1.05457x10^-34 y-2.84263x10^-78 y^2&lk=1&a=ClashPrefs_*Math-

 

[PeterDonis]

I modified the formula a little to look for energy change

Energy change of what? What physical process involving energy change are you trying to model? Until you answer that question, your efforts are not likely to be fruitful. I won't even comment on the rest of your post because I don't see what it's trying to model.

 

[mcjosep]

I suppose its modeling the time energy relationship between the energy of the attracting mass and the amount of time that is being dilated. I chose in the formula above to do one second minus x * Planck time and I did this to explore if you were to slow down time by a certain amount then would it have any effect on energy.

To be a little more clear, let's say we are on a tandem bike together (no reason its tandem except that you seem like a person that would enjoy that :) ) and we have a fly wheel on the bike. We are traveling at 10 meters per second and then I slow the bike down to 9 meters per second using a flywheel. So I slowed my speed down but picked up fly wheel energy.

That is exactly what I am trying to formulate. Except not the relationship between speed through space and flywheel energy, but rather I am calculating the relationship between speed through time and energy of the mass.

What got me thinking about this was imagining a cubic centimeter of time going slower by let's say for every one of its seconds we went by 50 years. Now let's say you were to drop a pencil on this little cube centimeter, and the pencils center of mass is not perfectly lined up with the cube. I believe the pencil would behave as if it hit a mass and bounce off to one side. Since mass is made of energy then I construed from this though that a change in time must be associated with a change in energy. Which then brought me to gravitational time dilation and this forum.

Let me know if this helps explain what I am trying to formulate.

 

[PeterDonis]

I suppose its modeling the time energy relationship between the energy of the attracting mass and the amount of time that is being dilated.

The energy of the attracting mass is just its mass. That doesn't change. The "amount of time that is being dilated" would just be the time dilation formula in terms of the mass (the first formula in your OP).

We are traveling at 10 meters per second and then I slow the bike down to 9 meters per second using a flywheel. So I slowed my speed down but picked up fly wheel energy.

This has nothing to do with gravity or time dilation.

I am calculating the relationship between speed through time and energy of the mass.

The only meaning I can assign to this is the time dilation formula (the first formula in your OP), as above.

imagining a cubic centimeter of time going slower by let's say for every one of its seconds we went by 50 years.

I'm not sure I understand what this means. Is this an object one cubic centimeter in volume, that is deep in the gravity well of a mass so that it is time dilated? Or is it just a cubic centimeter of empty space that is deep in the gravity well?

Now let's say you were to drop a pencil on this little cube centimeter, and the pencils center of mass is not perfectly lined up with the cube. I believe the pencil would behave as if it hit a mass and bounce off to one side.

If it's an object one cubic centimeter in volume, of course it would. If it's just a cubic centimeter of empty space, it wouldn't; the pencil would behave just as it would with any other cubic centimeter of empty space. Empty space is still empty space, even if it's deep in a gravity well.

 

[mcjosep]

The energy of the attracting mass is just its mass. That doesn't change. The "amount of time that is being dilated" would just be the time dilation formula in terms of the mass (the first formula in your OP).

ok I know you know this but I am calculating for E in E/c^2=m I replaced the mass with energy over c^2. that is how I am bringing energy into this.

This has nothing to do with gravity or time dilation

I know, I am just showing you how I am trying to compare time with energy in a simple way that may help you understand.

I'm not sure I understand what this means. Is this an object one cubic centimeter in volume, that is deep in the gravity well of a mass so that it is time dilated? Or is it just a cubic centimeter of empty space that is deep in the gravity well?

This is just a cubic centimeter of empty space in the palm of your hand right now, the only difference is within its boarders time clicks by incredibly slow. Use you imagination a bit.

If it's an object one cubic centimeter in volume, of course it would. If it's just a cubic centimeter of empty space, it wouldn't; the pencil would behave just as it would with any other cubic centimeter of empty space. Empty space is still empty space, even if it's deep in a gravity well.[/QUOTE]

Again use your imagination, even in Earth's gravity if something is accelerating/falling at 9.8 meters per second^2 and part of this something falls into a part of space where one second is really slow I would argue that it will not just simply fall go through it as if nothing is there. It is not in a gravity well.

 

[PeterDonis]

mcjosep, as a general note, you don't need to post in boldface unless there's something in particular you want to emphasize. There's no good reason to emphasize your entire post.

I am calculating for E in E/c^2=m I replaced the mass with energy over c^2. that is how I am bringing energy into this.

That still doesn't have anything to do with time dilation; it's just a unit conversion, from mass units to energy units.

I am just showing you how I am trying to compare time with energy in a simple way that may help you understand.

The flywheel example has nothing to do with "comparing time with energy". It's just shifting energy from one form to another.

if something is accelerating/falling at 9.8 meters per second^2 and part of this something falls into a part of space where one second is really slow I would argue that it will not just simply fall go through it as if nothing is there

You might argue that, but your argument is incorrect--at least, it's incorrect as a description of what GR says.

 

[mcjosep]

mcjosep, as a general note, you don't need to post in boldface unless there's something in particular you want to emphasize. There's no good reason to emphasize your entire post.

I was just making it bold and your part italics to show your part vs my part, I am not to savvy when it comes to make the quote blocks you use. I think I am getting better at it now.

The flywheel example has nothing to do with "comparing time with energy". It's just shifting energy from one form to another.

I guess in the example where the bike is having a decrease in speed and an increase in flywheel energy I was more trying to relate to my reasoning showing that with the decrease in the speed through time (similar to the bike with space), I am looking for an increase in energy somewhere else. I do not know how else to explain this. I'll try and think of something.

You might argue that, but your argument is incorrect--at least, it's incorrect as a description of what GR says.

I know that general relativity wouldn't allow for a cube a empty space having really slow time characteristics. Thats why it is just a thought. Do you know of a book or paper that would explain the effects of massive objects going through quick changes in time?

 

[PeterDonis]

I was more trying to relate to my reasoning showing that with the decrease in the speed through time (similar to the bike with space), I am looking for an increase in energy somewhere else.

I don't think this is a good analogy. The reason the flywheel speeds up in the bike example is dynamics: a force is exerted on the bike that slows it down and transfers energy and momentum from the bike to the flywheel. The reason a clock deep in a gravity well runs slow is geometry: the length of its path through spacetime is shorter than the length of a corresponding path much higher up. I think you'll be much better off trying to understand time dilation as a manifestation of spacetime geometry.

I know that general relativity wouldn't allow for a cube a empty space having really slow time characteristics.

Sure, it would, if the cube of empty space is deep in the gravity well of a large mass. Why do you think it wouldn't?