Galaxies in an Expanding Universe
By Richard J. Hanak, June 8, 2002
The Hubble relationship exhibits spherical symmetry. More concisely, Hubble's law is isotropic with respect to Earth. The isotropy of Hubble's law seems to confirm the philosophically attractive notions that when considered on a large enough scale, the universe is centerless, homogeneous, and isotropic about any location.
Let us assume that those notions are correct. If the universe is expanding, its expansion must also be centerless, homogeneous, and isotropic. An isotropically expanding system expands equally in all directions. Suppose that in some time interval the distance between two galaxies were to double. Isotropic expansion would require that the distances between all galaxies double in that same time interval. We can deduce, therefore, that distance ratios and angular relationships must remain constant during the isotropic expansion.
The light arriving from galaxies is our only source for information about them. In the present context we can consider light to include any kind of electromagnetic radiation that originated in a galaxy. Distances are calculated from the intensity of light, sometimes of one resolvable star, sometimes of the whole galaxy. Velocities are calculated from the shifts in wavelength in the spectrum of that light. Is square law the only cause of attenuation of the light intensity? Is the Doppler effect the only cause of red shift?
Hubble recognized that the radiation from the galaxies was emitted long ago. The information the light from galaxies carries to us can only be information about the galaxies as they were and where they were at the time they emitted that light. Expansion takes place during the passage of time. For that reason, time must be taken into account. Let us assume as initial simplifications that mutual gravitation has been negligible, that galaxy velocities have remained constant during the whole time of expansion, and that the velocity of light also remained constant. Adding an early inflationary phase and the decelerating effect of gravitation will not alter the outcome of what follows. For ease of illustration let us assume the Earth and five galaxies equidistant from each other in a ten billion years old universe.
The accompanying diagram (Galaxies in an Isotropically Expanding Universe) presents time vs. distance relationships and velocity vs. distance relationships for an expanding universe. Consider the upper portion of the diagram. The left-hand vertical axis represents time elapsed in billions of years since the beginning of the expansion. The lower horizontal axis represents the distances of galaxies from Earth, measured in light-years. On the top horizontal line of the diagram, the present time, are shown the Earth and five galaxies, equidistant from each other, with their respective velocities. The velocity of galaxy #5, the velocity of light, is shown as a limiting condition.
The lines connecting the present positions of the galaxies to the beginning of the expansion show the positions of the galaxies relative to each other and to the present position of earth at any time in the history of the universe. In order for light from a galaxy to arrive here now, the time it took for its light to travel here has to equal its distance from earth when it emitted that light. For example, five billion years to travel a distance of 5 billion light-years. In the diagram the line for present arrival of light shows this time-distance relationship. Please notice that the oldest light that can possibly reach us now is only half as old as the universe. To observe beyond that limit would require galaxy velocities greater than the velocity of light.
The intersections of the galaxy position lines with the present arrival line show the non-simultaneous distances from which their light was emitted. The vertical dashed lines projected down through the distance axis from those intersections show that distance measurements from arriving light cannot yield equal distances between our galaxies.
Let us now direct our attention to the lower portion of the diagram. The left-hand vertical axis represents velocities (in light-years per year) of our galaxies relative to earth. The horizontal axis above, as before, is distance from Earth’s present position. A curved line (labeled Observable Velocity vs. Distance for Isotropically Expanding Universe) depicts the relationship we should expect between observable distances and velocities of our galaxies. The straight line for Hubble’s Law, shown at the bottom of the diagram, cannot represent observed velocity-distance relationships in an isotropically expanding universe.
Now we can arrive at some conclusions. If the velocity-distance relationship (Hubble’s law) calculated from observed data is linear, the data cannot represent recession velocities in an isotropically expanding universe. If the data do not represent recession velocities, the universe has not been expanding and the observed red shift was caused by something other than the optical Doppler effect. If the universe has not been expanding, any theories requiring such expansion (or contraction) are somehow in error.