The answer below is all well though both it and you are suggesting that there must be 'space' or dimension containing areas outside our 'universe'.
Im sure in school you were told to picture the expansion in terms of a balloon. Hubble discovered that all distant galaxies were moving away from us (in every direction). He hypothesised that the distance between the galaxy and the earth is proportional to the velocity in which they are moving away (relative to the earth). This of course allowed him to define the arbitrary constant known as Hubble's constant.
Back to the balloon, imagine that you draw dots equally spaced around the balloon and label a central dot (for ease) as earth. Now begin to add air to the balloon and notice that the distance does not increase equally between the dots (this being comparable to galaxies). This is because every tiny area of the balloon has increased by a small amount adding to the new surface area, every minute fraction on the surface area on the balloon has increased equally but because there is a greater area between the further dots/galaxies (hence greater amount of small amounts [delta A]), the overall distance travelled is greater. Because displacement(s)=Velocity*Time and time is constant it means that if the distance increases, the velocity must have increased ergo Hubbles (VERY BASIC) model. Now in real life there are 3 dimensions in this expansion rather than the 2 within the balloon model but it does help grasp the concept.
Of course this wasn't your question, just it helps to have a basic understanding of what is happening incase you didn't at all.
In terms of your question; when we think of expansion, by definition we assume the object is becoming larger. But now think about RELATIVE expansion, what if everything that ever was - i.e. our universe (all the matter/antimatter=0 after taking into account the half pairs that are destroyed via black holes after random fluctuations bring the matter/antimatter pairs into existence) - was situated in a single infinitely small point in a non dimensionless (hard to grasp) 'place'. The fundamental component of everything is energy (EVERYT MASS contains it) and a fundamental law we know that the net energy in the universe is 0.
So now we've constructed the image of a nothingness (or tried to). That nothingness is our universe. composed of infinite 1s and -1s (matter and antimatter) adding up to the net 0 or nothingness of it.
Inside that nothingness is our universe which contains everything but is still non existant due to the net energy/matter amounting to 0. Now we can assume that the 'nothingness' has an arbitrary 'space' in which everything is situated. To explain this I propose you assume our universe is situated within the confines of this zero> 0. The space between the black line which defines the number is the universe. At the 'beginning' the matter within the universe (remember that the total=0 but within the confines of the 0 we can separate the 0 to = -1+1-1+1-1+1..... infinitely.) took up the entirety of the area between the black with no 'space' for anything else (i.e. the infinitely small, dense mass as proposed by the big bang model). But as the 'time dimension' progressed every tiny -1 and 1 within the confines of the 0 begin to shrink at an equal pace.
Now every point (some called 1 for matter or -1 for antimatter) has begun to shrink at an equal pace, try to imagine the space between them, remember the overall area within the 0 is EXACTLY the same but RELATIVE to the points (-1 and 1s) the 'space' appears to be expanding and as with the balloon model the further the points from each other, the greater the apparent area between them over a period of time (directly proportional to velocity i.e. every increasing velocity).
Now the problem with that model is FORCES, it does not account for them but assumes that as 2 or more 'points' (the -1s or 1s i spoke of) interact directly with each other they become regarded as a single point. So for this model we can assume that a galaxy is a 'point' of positive (+1) matter.
I hope this made sense to you, it's a pretty hard thing to explain as its so counter intuitive and almost philosophical.
Sources: I study geophysics at university and college - Though I am not a professor or at a similar level yet so don't expect 100% excellence in my answer.
Posted 476 day ago