Gravitational Time Dilation Due to Expansion of the Universe in 4th Dimension

Below post is a back-up of my posts in my Quora blog. I take a copy of them to here in case if something goes wrong in there.

(If you somehow have ended up on this page without any background knowledge, before continue, you better know that all below scenario is based on the idea that our universe is expanding in time -4th dimension- direction with speed of light. You can check the previous 2 posts of: 1. Do We Travel in Time Direction With Speed Of Light and 2. Expansion of the Universe as Travel in Time Direction if you are willing to understand this concept)

In previous posts, I have tried to explain a model which our universe expands with the speed of light in the 4th dimension direction and the speed of light was the universal speed limit. In that universe, within 3 spatial dimensions, we can only observe the stretching of the universe which we call the expansion today, while the real expansion is happening in time direction with the speed of light. We did not have any other knowledge in our hand except that the gravity is the dominant force in galaxies and universe doesn’t expand according to our observations where the gravity is dominant. But as I said; the expansion which they call today, corresponds to stretching in our expanding universe model. Now, we will apply this idea to our universe model and see what happens.

Do you remember below graph from the 2nd post?

We have used this graph to explain how the distance between objects increases while they travel in time direction. We have called them objects but we did not give them any feature like mass or gravity since we didn’t know about the basics of such concepts. Calling those points as objects was nothing more than calling them “things”.

If we want to understand the basics of any idea, we need to go deep down to its lowest level, to its simplest form. And a point is the simplest form that we can define in our universe. It has 0 dimensions. No width, no length and no height. Even its existence is a miracle under such conditions. But they are the most useful tools for thought experiments like ours and we will use them as our base. So our A, B, C objects are our points, the smallest parts of our universe. Remember that they don’t have any mass or size yet.

I will not claim yet that I know what the cause of the expansion is and what is happening to the structure of the universe during that expansion. What I know is; all the points must spread away from a center towards every direction during an expansion (if the fabric of space is made of points, of course. If it is wave based, then it is another scenario. But I am only capable of dealing with points; therefore I will proceed with them). We will build our gravitational time dilation mechanism on this idea.

Now we can proceed. Let’s go back to the idea of gravity being dominant within the galaxies and universe is not expanding there. We don’t know what is gravity or mass but we can simply fit this idea into our model like that: If universe is not expanding (actually not stretching) where the gravity is dominant; it means that; the distance between the points in that region of the universe does not increase. That means, the points in that region are travelling with the speed of light on time direction with a fixed distance from each other as below:

This is very similar to our previous graph. We have 2 points which we call A and B on their blue observable 1D universe while that universe is expanding on time direction with speed of light which is indicated with the red arrows. This time, there is one difference, though. Instead of keep following their routes, A and B points somehow keep the distance between them and expand towards the black arrows direction instead of the red ones, just as the matter would do where the gravity is dominant in our universe. This is the only feature we can appoint to matter right now since we don’t know any other features of it. And this feature results a slight deviation from the previous route of our points.

After t’-t amount of time, A point reaches to A’’ instead of A’ and B point reaches to B’’ instead of B’. We don’t know what did cause them to move together or what the distance from each other was when they have started their joint movement first. But one thing we know for sure; the stretching rate of our universe. That means, 2 points with 1 mega parsec distance from each other should move apart on x direction by ~70,8 km after 1 second. If we assume 2 points with that much of distance apart from each other starts to move keeping their distance between them fixed, we can check what will happen to their speed in time direction. If |BB’’| is longer than |BB’|, covering that distance in 1 second will mean that they are travelling faster than the speed of light which is not possible. So their speed in 4th dimension direction should decrease accordingly to keep that speed within the speed of light limit.

In order to calculate |BB’’|, we can neglect the curvature of our universe (green arc) and treat our BB’B’’ shape as a right triangle and use Pythagorean Theorem since the curvature will be very small considering size of the universe:

|BB’’|² = |BB’|² + |B’B’’|²

|BB’’|² = 299792,458² + 35,4²

|BB’’| =~ 299792,460 km

It is a very small difference (about 2 m/sec) but it anyway exceeds the speed of light. Therefore, our objects must be subjected to a time dilation and relative to other points in the universe which doesn’t jointly move, their 1 second shall be different than the others. If we wanted to find the amount of time dilation we could simply divide |BB’’| to |BB’’’|. We will come to that soon, but first let’s take a broader look on where we are right now.

As a summary; our points were first supposed to end up on A’ and B’, then they have started to move jointly and they were supposed to end up on A’’ and B’’, then they have been subjected to a time dilation due to their joint speed and they have ended up on A’’’ and B’’’. Final status is something like in the below sketch.

Now, we are sure that there must be a time dilation. But this calculation is only based on the points which are 1 MegaParsec apart from each other. What if our points had a different distance between them? It is better to leave that distance as a variable and write a formula for the amount of time dilation depending on the distance between 2 points. In order to do that, let’s find out a formula where our previous 35,4 km |B’B’’| figure is the variable L distance between 2 points.

|B’B’’| = (Expansion Rate /2) x L / 1 Megaparsec

|B’B’’| = 35,4 x L / 3,1 x 10^19

|B’B’’| =~ 1,47 x (10^-18) x L

Now we know the length of |B’B’’| as a function of L. So, we can calculate |BB’’| from Pythagorean Theorem and reach to time dilation amount after that.

|BB’’|² = |BB’|² + |B’B’’|²

|BB’’|² = 299792,458² + (1,47 x (10^-18) x L)²

|BB’’| = (299792,458² + 1,3152 x 10-36 x L²)^(1/2)

And the time dilation shall be calculated as:

t’/t’’’ = |BB’’| / |BB’’’|

t’/t’’’ =~ (299792,458² + 1,3152 x 10-36 x L²)^(1/2) / 292792,458

As a summary, our A and B points which were travelling in time direction with speed of light on their own routes were supposed to end up on A’ and B’ locations after 1 second due to expansion of the universe in 4th dimension direction. But instead of following their routes defined by the expansion, they somehow have started to keep their distance between them and started to follow a new joint route. That should have cause them to end up on A’’ and B’’. But to reach A’’ and B’’ in 1 second, they should be travelling with a little bit more than the speed of light. So instead, their speed in new joint direction have decreased to speed of light and caused a time dilation.

Our formula shows us that; if the L between 2 points is not a big amount, the time dilation effect will be really, really small. In order to have a considerable gravitational time dilation effect, there should be a lot of joint points moving together on the same direction at the same time. Since I don’t know any details about the structure of space or the finest elements of the matter, I will not attempt any calculation on the gravitational time dilation effects of such bigger structures. It should be also noted that since the above formula is only related with the distance between points, it means it only deals with the density of 2 space points. To see the gravitational time dilation effects on multiple jointly moving points, the formulation shall be improved to deal with more than 2 points. I suspect that the effect will not be increasing linearly with the number of points but will probably be increasing exponentially. Anyway, I will not go into that detail right now.

What I have tried to explain in this post was; how the gravitational time dilation concept can be achieved in our expanding universe model with a simple idea like jointly moving points. It is also possible to reach the concept of gravity with jointly moving points idea based on the same model and the next post will probably be about that subject if I can warp my head around it well.