NOTE: I have made a significant revision to this answer and the conclusions are different to the previous version.
Or is the space actually something that wants to expand everywhere and
unless the two particles are bound by some force, the expansion will
kick in and they will slowly start separating again?
As explained below, even in the expanding spacetime model, there is no drag on objects due to the Hubble flow. The two objects will not start separating again. As Sten pointed out in his answer they will tend to drift together again due to the gravity of matter between them in a non empty universe. The expanding spacetime concept does mean that massive objects at rest with bulk flow will experience no time dilation and that the worldlines of photons are different in the two models. In the expanding spacetime time model the redshift is attributed to the 'stretching' of photons as they travel through the expanding space. In the kinematic model the redshift is due to relativistic Doppler shift and to a certain extent there is an opposing blue shift due to gravitational potential in a non empty universe. Common analogies such as transparent disks stuck on the surface of an inflating rubber balloon, are misleading. A common misconception is that objects have to be 'locally bound' to resist the expansion. Consider an exaggerated scenario for visual effect. imagine we tether a large object somewhere far outside or galaxy where the recession velocity is 0.5c to something fairly stationary inside our galaxy and cut the tether. What happens if we cut the tether? To first order both models predict nothing! There will however be slight inward drift due to gravitational effects. As Peacock explains in his paper, we can imagine a sphere of low density matter enclosing one object and the other object is on the surface of that enclosing sphere. There is a gravitational potential due to that enclosed matter. We can consider the gravitational attraction of the total mass in the enclosed sphere to be equivalent to the same mass concentrated as a point at the centre of the sphere as per Newton's shell theorem.
Wikipedia states "In reality, the expansion of the universe can alternatively be thought of as corresponding only to the inertial motion of objects away from one another."
The same Wikipedia article also states "Contrary to common misconception, it is equally valid to adopt a description in which space does not expand and objects simply move apart while under the influence of their mutual gravity.{2}{3}{4}"
{2} Peacock (2008), arXiv:0809.4573
{3} Bunn & Hogg, American Journal of Physics 77, pp. 688–694 (2009), arXiv:0808.1081
{4} Lewis, Australian Physics 53(3), pp. 95–100 (2016), arXiv:1605.08634 n
Let's compare two basic models, the SR ballistic model in flat spacetime (essentially the flying apart model) and a Hubble flow model based on the FLWR metric which assumes space is expanding. While in a matter dominated universe, there is an effect on observed redshift due to gravitational potential, I have deliberately ignored the effects of gravitation in order to isolate and focus on the effects on observed red shift due to kinematic effects.
Let's say we observe a very distant, high z, supernova event that appears to be 42 billion light years away from calculations based of its luminosity. The SR model always predicts a recession velocity of less than the speed of light so it has difficulty explaining how an object can be 42 billion billion light years away when the universe is thought to be only about 14 billion years old. This does not mean the ballistic SR model is fatally flawed, because what everyone forgets to take into account is relativistic beaming, which reduces the observed brightness of the object making it appear further away than it really is. Relativistic beaming explains why the object appears to be 42 billion light years away and appears to have a velocity of about 3c. The SR model also agrees with the HF model on observed Doppler time dilation, without any modification. This does not make the two models agree in every detail. There are subtle differences, as detailed below and the predicted distance vs redshift are slightly different as illustrated in the graph right at the end of this answer. This has consequences for whether the apparent acceleration of the expansion is just an artifact of using the expansion model. Of course we can also explain the observed distances, by giving the distant galaxy a head start of 28 billion light years during the inflation period just after the big bang, but this is a form of (extreme) space expansion in itself. I explore the differences of the two models in much greater detail below:
Another thing the SR flying apart model has difficulty explaining is the apparent accelerating expansion of the universe. However this recent paper that has been peer reviewed and published in a respectable journal casts serious doubt on the accelerating expansion hypothesis.
As requested by the OP, gravitational attraction between galaxies is treated as being negligible and accelerating expansion due to dark energy is ignored.
Basic definitions of the two models:
| Quantity |
SR Ballistic Model |
Hubble Flow Model |
| Time dilation |
A distant galaxy is subject to time dilation as a function of its velocity relative to us as per SR. Even objects conforming to the Hubble velocity distance relationship are subject to time dilation. |
A distant galaxy is only subject to time dilation if it has velocity relative to the bulk flow (local or peculiar velocity) |
| Bulk flow Recession Velocity |
$v_r = \frac{z(z+2)}{z(z+2)+2}$ |
$ v_r = Log(1+z)$ |
| Maximum velocity |
Speed of light (c) |
< Infinite |
| Distance when emitted |
$D_e = \frac{T_{Now} \ z(z+2)}{2z(z+2)+2}$ |
$D_e = \frac{T_{now} \ Log(1+z)}{(1+z)}$ |
| Light travel time |
$T_{ltt} = D_e/c$ |
$T_{ltt} = D_e(e^{v_r}-1)/v_r $ |
| Distance (Lum) |
$D_{lum} = D_e \frac{(1+v_r)^{(3/2-\alpha/2)}}{(1-v_r^2)^{(3/4-\alpha/4)}}$ |
$D_{lum} = D_e (1+z)$ |
| Apparent recession velocity |
$V_a = D_{lum}/T_{now}$ |
$V_a = D_{lum}/T_{now}$ |
An alternative expression for $D_{Lum}$ in the Hubble flow model is $ Log(1+z)T_{now} \quad$.
The SR luminosity equation is complicated because it has to take into account the effect of relativistic beaming on the apparent luminosity of an object. It is something that must be taken into account, because it is an unavoidable prediction of special relativity. In the equation, $\alpha$ is the spectral index of the observed object. This has a value of between zero (for mostly high frequencies) and 2 (observed spectrum concentrated at low frequencies). $V_r$ is the observed recession velocity of a galaxy at rest in the bulk Hubble flow. The relativistic beaming effect that is completely ignored in all cosmology papers is a key factor in allowing the SR flying apart model to match observations.
In the above diagram assume a supernova event occurs at J and the proper time interval of this event is 20 days. (D to S). Doppler time dilation predicts a time dilation factor of (z+1), so a supernova that has z=1 should be seen over a period of 40 days. This has been confirmed by actual observations. For a model to be valid it must be able to reproduce this time dilation factor. In the above diagram the SR model is on the left and the HF model is on the right. Both models reproduce the (z+1) Doppler time dilation factor. For the SR model, the coordinate time interval of the event is 25 days due to time dilation. It also occurs at a later cosmological coordinate time in the SR model, but due to SR time dilation the proper age works out the same as in the expanding model. There is exact agreement between the two models in this comparison and the signals arrive at the same time. Where they differ slightly is in the predicted distance from the observer of the supernova event.
In the above diagram the dashed line represents a projectile in the SR Ballistic model that is launched at 0.5c. In this model its motion is entirely due to its initial momentum. If there is a galaxy also moving at 0.5c away from us that is at rest with the bulk Hubble flow (blue worldline) and initially at 50 units distance, the projectile will never catch up with distant galaxy and in fact will remain at 50 units distance away from it for all time in the SR model. Previously I incorrectly stated the HF model predicts the projectile will follow the trajectory represented by the solid red curve, but this was based on the assumption that a projectile with mass would comply with the same equation of motion as a photon. The massive projectile would in fact follow the dashed straight path in both models. A consequence of this fact is that in both models any inertially moving massive object will asymptotically come to rest locally in the Hubble flow.
The reason a light particle behave differently to a massive particle is that it has to maintain a local peculiar velocity of c and loses energy by changing to a lower frequency or longer wavelength. The massive particle on the other hand loses energy by slowing down it peculiar velocity. Similar behaviour can be seen in the Schwarzschild metric. A projectile fired upwards slows down locally and eventually comes to a stop. A photon climbing out of the gravitational well always has a local velocity of c but loses energy by changing wavelength instead.
The x coordinate of a photon moving outwards in the same direction as the Hubble flow as a function of time in the expanding universe model is given by:
$x_{out} = \frac{c}{V_h} \ \ln\left(1+\frac{V_h t}{D_e}\right) (D_e+V_h t)$
where $V_h$ is now the local Hubble flow velocity of a target galaxy and $D_e$ represents the distance of the target galaxy at the time the photon left. The photon trajectory would be more like the solid red curve in the above graph.
As mentioned at the start and as discussed above, inertial objects will move in exactly the same way in both models. Two objects that are initially at rest with respect to each other and initially comoving will maintain constant spatial separation as illustrated in the diagram below by the dashed purple trajectories. However it can be noted that over time the separation of the two objects appears to be getting smaller, because while they maintain constant separation relative to each other surrounding galaxies are moving apart relative to them.
In the next chart, I have plotted the luminosity distances against observed redshift for both models. The orange curve is the luminosity distance according to the SR Ballistic model and the blue curve is the distance according to the HF model. I have assumed a value of $\alpha = 1.6$ for the above plot as this value gives distances in the same ball park as each other. In this answer I have not taken a position on which model is the correct model. However, IFF the SR ballistic model is the correct model, using the Hubble flow model would incorrectly predict that low to medium z objects are too far away and high z objects are too close, creating an illusion of accelerating expansion.
Wolfram equation and plot here.
