Time dependent gravity exploration

Time dependent gravity exploration

In our universe we have a gravitational constant – i.e gravity is not dependent on time.  If gravity changed with respect to time then the gravitational force exerted by the Sun on Earth would lessen (or increase) over time with all other factors remaining the same.

Interestingly time-dependent gravity was first explored by Dirac and some physicists have tried to incorporate time dependent gravity into cosmological models.  As yet we have no proof that gravity is not constant, but let’s imagine a university where it is dependent on time.

What if gravity is time dependent?

The standard models for cosmology uses G, where G is the gravitational constant.  This fixes the gravitational force as a constant.

However, having time dependent gravity could be one way of explaining the Big Bang.  If gravity varies such that in the first few moments after t = 0, gravity was negative and thus repulsive [and close to infinitely strong], this could explain the initial incredible universal expansion before “regular” attractive gravity kicked in (after say t = 1).

The gravitational constant, G has only been measured to 4 significant figures:

G = 6.674 x 10^-11m³kg^-1s^-2.

Therefore if there is a very small variation over time it is possible that we simply haven’t the accuracy to test this yet.

Making some model constraints

I’m going to fit some different models for time-dependent gravity – one in which the change in gravity over time is increasing and approaching a limit and one in which gravity is decreasing to a limit

For the 2 models I want to fix certain points

(a) I want to have an asymptote at t = 0 such that the graph approaches negative infinity as t approaches 0. This will give a close to infinite repulsion in the initial stages after t=0 to provide the thrust of the Big Bang

(b) I want to have gravity become positive (and therefore attractive) after a very short time – let’s say after t =1 second.

(c) I want to have the gravitational constant G to be 6.674 x 10^-11m³kg^-1s^-2 after the current age of the universe (13.7 billion years) to match current observations.

Model 1:  Gravity increasing and approaching a limit

What we want is something that looks like this:

This would give us the initial negative gravity, followed by gravity strengthening towards a limit (b).  If we were at some point (c) that was close to this limit then G may feel constant as the variation over time would be very small.

This graph can be created by considering a reciprocal graph (y = 1/t) in which we then reflect in the t axis, add a horizontal stretch factor 1/k and translate upwards by p:

Now let’s set the horizontal asymptote as 6.6744 – a number close to our current measurement of G. This would give:

Next we can make sure this passes through (1,0):

And then lastly we can check that this model would give our current value of G for the current age of the universe.  For this we need to change the age of the universe to seconds as our time is measured in seconds.

and then put this into our equation:

We can see that this does match what we would currently measure.

Model 2:  Gravity decreasing and approaching a limit

This time we want something that looks like this:

This time we can use a function of the form:

Now let’s again set the horizontal asymptote as 6.6744.  This would give:

As last time we make sure that the graph goes through (1,0):

This then gives the function:

We can check as last time that this gives the measured value of G at the current age of the universe:

This model would give a maximum gravitational constant of 6.675 which then approaches 6.6744.

Does this fit the universe?

Once we have our models, the next question would be what would happen if these models were correct?  What would the consequence be of a universe where gravity was increasing to a limit or decreasing to a limit?  We’ve seen that both of these models provide a possible mechanism for the Big Bang and fit current measured values of G – but this does not therefore mean that these are correct!  

There has been serious study into whether time dependent gravity is consistent with our model of the universe.  For example – would a gravity decreasing over time still allow for universal expansion (which we see)?  I’ve previously written a pretty heavy-duty maths post on this here.  The summary is that you can show mathematically that even in a universe where gravity is time dependent (and decreasing), we would still be able to have an accelerating universe like we see today.  

What other models could fit our data?  A universe where gravity increases without bound?  A universe where gravity tends to 0?  Have a play!