### Introduction

Hubble's Law describes a (linear) relation between the distance (d) of a galaxy and the (radial) speed (v) of the galaxy, expressed as: v = H * d , with H being the Hubble Constant
In this document this is called: the second Hubble's Law

The purpose of this document is a critical evaluation to what extend this law (the second) is true. Emphasis is on the relation between v and z.

For more information about Hubble's law please study the documents in the Literature List. Some of those documents are referenced in the following paragraphs

### Critical evaluation

Hubble's Law is about two subjects:
1. The distance of a galaxy and the size of the Universe. This is the static part. In this document this is called: the first Hubble's Law . In fact this is the relation discovered by Edwin Hubble
2. The speed of a galaxy and the expansion of the universe. This is the dynamic part, i.e. the second Hubble's Law
Those two subjects are intertwined, but in fact they can be studied separately.

To derive the first law from the second law is easy. The speed of a galaxy is determined by studying its redshift and the following formula applies: v = c * z. With z being measured and equal to: z = d labda / labda.
As such we get v = H * d = c * z. Combining the two parts we get: z = (H/C) *d or d = (c/H) * z. This is the first Hubble's Law
The constant H/C is obtained by observing galaxies of which both d and z independent can be established. By plotting those points in a x,y diagram and by using a least square fitting method the factor H/C can be calculated.
The first Hubble's Law can than be used in cases where only the factor z is observed in order to calculate the distance d of a galaxy.
IMO the factor H/C should have been called the Hubble constant H'. That is the factor that Edwin Hubble, based on his observations of z and d, calculated.
What is important is that formula D = z / H' is independent of c and can be used when z > 1 !

For the speed v we get than: v = c * z = c * H' * d. This is the second Hubble's Law

To calculate the speed of a galaxy is a whole different subject. The problem is partly addressed in document (5) above where in paragraph "Observability of parameters" we can read:

Strictly speaking, neither v nor D in the formula are directly observable, because they are properties now of a galaxy, whereas our observations refer to the galaxy in the past, at the time that the light we currently see left it.
In short what we measure (if we can) is a distance (position) and speed in the past while what we want is the distance (position) and speed now.
That is the problem!
• This is not a problem if we consider our Solar system, because we can perform multiple observations of the positions of the Sun and the planets over a long period of time. Using Newton's law and taking those positions into account we first calculate the masses of those objects, then the positions and velocities of a certain moment t0 (within this time period) and finally the positions at a moment tn (i.e. the present)
• This is also not a problem for the stars in our Solar system. We can calculate the speed v of a star by different observations and we can establish that there a linear relation exist between v and z, but we cannot establish the linear relation between v and d. See also: The Motions of the Stars. The document mentions that what we want to know are the "True Space Motions". That is part of the problem. What we want to know are the positions and motions of the present based on measurements which represent the past.
• The problem starts when you consider our nearest galaxy the Andromeda galaxy M31 or NGC 224 At that distance you cannot measure the speed of individual stars. You can only observe based on redshift that the stars of certain regions are approaching us or are receding from us. Anyway at that relative short distance Hubble's Law (First nor Second) does not apply.
To quantify: Based on H=74 km/sec/Mpc and a distance of 2,5 Mly we get a receding speed of roughly 182/3,26 = 56 km/sec, which is rather high compared to the true speed which is roughly 100 to 140 km/sec towards the Milky Way galaxy
Which means that neither the first (distance) nor the second (speed) Hubble Law are applicable. (z=0.0006)
• The question now becomes at what distance are both law applicable.

### The problem with Hubble's Law

What are the problems with Hubble's Law ? In fact there are three problems:
1. The linear relation between redshift (z) and distance. (First Hubble's Law)
2. The linear relation between speed and distance. (Second Hubble's Law)
3. The size of the Universe
The following sketch explains the issues involved:
 ``` ^ ^ * | 1 | * * | | * * | z | * * | | * * | | * <---------------* 0 *--------------> 1 distance = d speed = v 1 figure 1 ```
• The left side shows the linear relation between distance d and redshift z. This is what is called the first Hubble's Law.
The question is: is this Law valid at all ranges. The answer is definitively No at short ranges i.e. at the scale of the Andromeda galaxy, because here we measure a negative value for z.
• The right side shows the linear relation between redshift z and distance v. This is the doppler shift formula.
• If you combine both figures you get the second Hubble's Law showing a linear relation between distance d and speed v.

### The distance versus redshift (z) relation

The distance versus redshift (z) relation comes in two flavours:
• First we can calculate the distance by measuring parallax. There are two major problems with this method.
• At short ranges we can measure parallax accurately, but the linear relation between z and d does not exist. In fact we use there z as a function of speed. This is done to measure the speed of the Andromeda Galaxy. See above.
• At larger distances parallax based methods are very difficult because the angles involved are becoming very small.
• Parallax methods use the direction of light. Light is bended by mass. That means they become more inaccurate over large distances.
• The second method is by using Luminosity or Magnitude. There are also certain problems with this method.
• At short distances the same as with parallax i.e. there exists no linear relation.
• At larger distances the same problem exists.
See for example (at the bottom) of page 2 in document (13) which reads:
Unfortunately, NGC 4258 is too close to determine H0 directly (i.e. by dividing its recessional velocity of 475 km s-1 by its distance), since the galaxy’s deviation from the Hubble flow could be a significant fraction of its recessional velocity.
The subject of the sentence is not velocity but redshift.
• Document (9) figure 12 at page 53 shows the linear relation between cz (i.e. the speed) and Magnitude (distance) based on Cepheid variable stars. This figure shows is a linear relation between z and Magnitude. In short the Hubble constant H or the first Hubble's Law or the left part of the above sketch. You get such a relation if both parameters are dependent of each other. They are because both are properties of the same physical system i.e. light. This becomes more of an issue when space expansion is involved.
• Figure 12 is derived from Cepheid stars. Those stars show a periodic behaviour. The duration is also stretched (Time dilation)

A whole different parameter to calculate distance is parallax. In fact parallax of NGC 4258 is used as an anchor point to calibrate this figure. For more details See Document (10)
The same document also shows that NGC 4258 is receding at a speed of 537 km/sec. That value is calculated by using v = c * z with z being observed. The distance d = 7.16 Mpc (11) is calculated by using c * z = H * d or d = c * z / H with H = 74 km/sec/Mpc.
For UGC 3789 (See Document (13)) the same method is followed. The speed v = 3325 km/sec is calculated by using v = c * z with z being observed. The distance d = 46 Mpc is obtained by dividing c * z by H with H = 74 km/sec/Mpc.
For NGC 6323 See document (14). The speed is 7772 km/sec. H = 72 km/sec/Mpc. Dividing those 2 numbers gives a distance of 110 MPC.

Dividing the speed for each of these galaxies by c then the redshift value z for NGC 4258 = 0.002, for UGC 3789 = 0.011 and for NGC 6323 = 0.026. In fact the redshift values are the "observed" values. (This is not 100 % true. See at the top of this document)

In the next paragraph I will explain that those calculated speeds most probably are not physical realistic.

### The redshift (z) versus velocity (v) relation

For the redshift (z) versus velocity (v) relation there are three issues IMO:
1. At short distances the relation tells you something about the speed v of the source.
2. At large distances the relation tells you something about the expansion of space between emission and receiver.
3. The question is: Does this relation at large distances also tells you something about the speed of the source?
IMO the answer on the last question is: No. (Document (16) shows that the whole issue is not so simple)
The reason is explained in the following sketch:
 ```t4 z 1 2 3 .G-------> v4 | 1 2 3 . d4 | 1 2 3 . t3| 1 2 3 .------> v3 | 1 2 3 . | 1 2 3 . t2| 1 2 3 .-----> v2 | 1 23 . | 1 23 . t1| 1 3.----> v1 0-------------------G--------- distance d1 Figure 2 ```
Figure 2 shows:
• A galaxy G at t1, a distance d1 away from the Observer moving away from the origin with a speed V1.
• The same galaxy G at present position t4, but now at a different distance d4 and with a different speed v4.
• The path of the Galaxy between t1 and t4
• Light emitted from the star at t1.
In fact the sketch shows three light paths (frequencies) identified as 1,2,and 3 assuming that the star emits only one frequency (Is a "laser")
1. Light (line 1) emitted at t1 from a certain frequency f1. Light has a speed c and moves in a straight line towards the Observer and reaches the Observer at t4.
2. Light (line 2) emitted at t1 but with a frequency f2. Line 2 shows a frequency shift as a result of the speed v1 of the galaxy. Line 2 runs parallel with line 1
3. Light (line 3) emitted at t1 but with a variable frequency from f2 to f3. Line 3 shows a frequency shift caused by both the speed v1 of at t1 and expansion of space between t1 and t4.
• The sketch does not shown:
• The situation for M31 or Andromeda Galaxy, because M31 has a speed v1 towards the Observer. In the case of M31 line 2 should be drawn left from line 1. Also line 3 should be drawn left from line 1 and falls in between line 2 and line 1
• The movement of the observer. In fact the Sun moves around the centre of our Galaxy with a speed of roughly 220 km/sec and this has to be taken into account For M31, NGC 4258
• The sketch shows a linear relation between the distance of the Galaxy (at t1) and the Observer (at t4) when you remove the speed of the Galaxy at t1. In fact this is the distance between line 2 and line 3. This shift is caused by the expansion of space, however it is not sure if this relation truly exists.
To calculate that relation is difficult for Galaxy NGC 4258, because v1 can not be calculated directly. You could claim that v1 = 0 but than you could introduce a large error in z. You need an independent calculation of the distance at t1. To do that, you need a parallax measurement, but that is also not easy.
• The sketch shows that there exist a linear relation between z and v, with both v and z increasing The distance is even accelerating. The question is if that is accordingly to observations and or is this physical possible.

The answer on the first part is no. You cannot measure the present speed directly. What the sketch shows is more or less instantaneous action, but if that exists is highly questionable. Also what that sketch implies is that the present speed is increasing and can have almost any value, even much larger than c. This is highly questionable. You can solve this mathematically and reduce the speed, but at the other side this should also be physical realistic.

In fact also other situations are possible. IMO it is much more acceptable that the average speeds of the Galaxies are decreasing as the Universe evolves. IMO the following figure is much more physical acceptable:

 ```t4 z . 1 2 3 .G-> v4 | 1 2 3 . d4 | 1 2 3 . t3| 1 2 3 .--> v3 | 1 2 3 . | 1 2 3 . t2| 1 2 3 .---> v2 | 1 23 . | 1 23 . t1| 1 3.----> v1 0-------------------G--------- distance d1 Figure 3 ```
Figure 3 is based around two concepts:
1. That space expansion is a local physical concept and not everywhere the same.
2. That the expansion of space decreases as the Universe becomes older.
That does not mean that the expansion of space has actual stopped and v4 = 0 (i.e. speed at present = 0)
The consequence is that the size of the (observable) Universe will be smaller.
What is also important, that in order to describe the evolution of the universe, no relativistic redshift is involved. Redshift basically only tell you something about the speed of the source. Redshift could tell you something about distance assuming that space expansion is every where the same along the line going from source to observer (and if the speed of the source is zero) In fact it is more one or the other. The more it tells about the distance the less about the speed.

### Universe

The problem with figure 3 is that the observer is the origin.
In the following sketch we observe the same as in figure 3 but from a different reference frame: the rest frame of the micro wave background radiation.

 ``` G2. z | .G1 v4 <-1 2 3 | t4 .-> v4 .1 2 3 | . d4 . 1 2 3 | . v3 <--. 1 2 3 | t3 .--> v3 . 1 2 3| . . 1 2 3 . v2 <---. 1 2 3 t2 .---> v2 . 1 23 . . | 1 23 . v1 <-----. t1 | 1 3.----> v1 -----X------G2-----0-----G1-------- d axis d4 . | . d1 . | . v0 <----------.---------->v0 t0 Figure 4 ```
Figure 4 shows:
1. The position and velocity of Galaxy G1 at t1. D1 is the distance from the origin for G1 at t1. At t1 light is transmitted from G1.
2. The position and velocity of Galaxy G2 at t4. D4 is the distance from the origin for G2 at t4. This point is identified with an X. Galaxy G2 harbours the observer. Light transmitted at t1 reaches the Observer at t4. The light path is the d axis and is identified with the numbers 1 assuming that the light consists of only one frequency. There are two more light paths drawn of different frequencies.
• Line 2 shows a frequency shift (redshift, z) caused by the relative speeds of both G1 and G2 at the moment of emission at t1. The further away both Galaxies at t1 the larger this influence
• Line 3 shows a frequency shift (redshift, z) caused by the expansion of space. This part is described by the Hubble
3. The sketch also shows the position of G1 at t4 also the position of G2 at t1. Those positions are in reality most probably not along the d axis.
4. t0 is the moment of the Big Bang. What this sketch suggests is that galaxies at large distances (like G1) could be much older as currently assumed.

### Hubble Constant

Is the Hubble constant a constant?
Is the Hubble constant defined as the relation between redshift and distance as in the first Hubble's Law a constant?
Locally the answer could be: Yes.
However what is the answer for large distances ? Over a large time period ? Over a time period comparable to half the time between now and the Big Bang ?
IMO the answer depends how space expansion influences (shifts) light frequencies and if this shift always was and is the same. I doubt that. IMO shortly after the Big Bang this shift was much larger than at present.
What this means is that the relation between distance and redshift is time dependent.
The question is how do you demonstrate that. The problem is you cannot. You cannot measure parallax at those large distances.

### The Redshift-Distance and Velocity-Distance Laws. Document (15)

In this document at page 29 we read
Eddington said (1930):"it is a though they were embedded in the surface of a rubber balloon which is steadily inflated." An expanding rubber surface aptly illustrates some of the properties of curved and dynamic space.
I have doubts. See above.
In an expanding homogeneous and isotropic space, let comoving markers A,B,C,D be equally spaced in a straight line. Homogeneity requires that if B recedes from A at a velocity v (was alpha * H(t)) then C simultaneous recedes from B at a velocity v and so on. Hence c recedes from A at a velocity 2 * v, D at a velocity of 3 * v and so on.
That is mathematical 100% true. But is this also true in reality? How do you prove this by experiment for all distances? What this description implies that the density in the total Universe is and stays everywhere the same (while locally this is not the case). In fact what you have to demonstrate is that the present Universe IS homogenous i.e. that the density IS everywhere the same. A prove implies that you do not use the law by itself to demonstrate this. As said above such a law requires instantaneous communication.
Light emitted toward the observer by a body outside the Hubble sphere travels in space at velocity c, but because space itself recedes superluminally, the light actually recedes. The light may eventually reach the observer, however, if the Hubble sphere expands in the comoving frame i.e. dLH/dt > c.
The last can be explained much simpler. Assume that light is emitted at a distance d on the Hubble sphere. This light is at rest relative to the Observer. Space expansion is not a constant phenomena but decreases over time. This means that the Hubble sphere increases implying that the a point at a distance d on the Hubble sphere, slowly moves inside the Hubble sphere. The consequence is that the light slowly starts to moves toward the Observer.
IMO this is a completely theoretical problem. A much more practical problem is the distance between the two points, one from which light was emitted (that we see now) and second from our present position, versus the age of the Universe.
Wave stretching illustrates an important application of the expanding space paradigm
It is the other way around: Wave stretching comes first i.e. Expanding space is the explanation of wave stretching. If this is true is a matter of discussion.

### Balloon model - Classical

In the Classical balloon the surface of the balloon is studied while the balloon is blown up. This represents a 2D Example.
The study starts by drawing three objects on the surface of the balloon. What the study shows is that the distance between each of those points (galaxies) increases every where with the same ratio.
That is correct, but is this a good analogy to explain what happens in the Universe in total?
• The first problem is that you should not study what happens axial (along the surface) but both axial and radial (line of sight)
• The second problem is that this model leads to the formula: v = omega * r. With v being the axial speed, while what you want is v = H*d with v being the radial speed.
• The third problem if you assume that the formula v=H*d is correct, with v being the radial speed and d being the proper distant than you assume a physical condition which requires instantaneous action over a distance millions of lightyears away.

### Balloon model

In order to explain the Big Bang and Hubble's Law often a balloon is used. Here I will do the same.
In the classical example we study what happens at the surface of the balloon, when the balloon expands.
In this case the balloon represents one layer inside the expanding Universe at a certain instant after the Big Bang. The total Universe is much larger as the balloon.
Consider a balloon which we slowly blow up starting at t0. Two specific moments are considered as shown in the following picture:
 ``` 3 . |+ . . | + . . | + . . 1 + . . . | .+ . . . | + . blowing . . | + . ----> M------M------O------2------4 direction . . | . . . . | . . . . | . . . . . . | . . | . . | . . Figure 5 ```
Figure 5 shows two circles. Each circle shows the surface of the balloon at a certain instant. The inner circle at t1. The outer circle at t2. The mouth piece is the letter M.
• The letter O is the moment t0 that we start blowing. This is the moment of the Big Bang..
• At t1 we place one finger of our left hand at the left side of the balloon. This is point 1. This point represents us, the Milky at an early stage. You should keep your finger at that position.
• At t1 we place one finger of our right hand at the front of the balloon. This is point 2. This point represents a far away Galaxy at an early stage. You can remove your finger, but you should remember that position because it will disappear inside the balloon.
• At t2 our left finger has moved to point 3. This point represents our present position.
• At t2 the far away galaxy has moved to position 4, but that is not so important.
What is important that at t2 we will see the far away galaxy from t1. The light from that galaxy has followed the line going from point 2 to point 3 identified with the + sign. At t2 we will also measure a certain z value (redshift):
• This value is a combination of the relative speed between the two galaxies at t1 and the continuous expansion of space between the two points 2 and 3 (which implies distance)
• The value of z is not a function of the present speed of the far away galaxy which presently is at position 4

An important picture to study is the following: Astronomy Picture of the Day which shows galaxies in the past at all distances.

• IMO the most important message is, that not all galaxies are the same: The most distant galaxies, the faintest galaxies, the galaxies with the high z values are the least evolved compared with the nearest galaxies. What this picture demonstrates is an evolution in galaxy structures.
• What this picture would demonstrate is that the Universe is not homogeneous, assuming that what we see is the present situation.
• On the other hand, because what we see/observe is the past this picture does not demonstrate that the total Universe is homogeneous
• At the same time, assuming different expansion speeds (compared to the lineair law described above) there could be other processes involved which take care that the Universe stays uniform.

### Conclusion

Figure 3 uses the concept that the second Hubble's Law i.e. v = H * d or v = c * H' * d is not valid. That means based on the measurement of z or d you can not make any prediction about the present speed and position of the Galaxy involved. What Figure 3 also tells you that based on the measurement of z you have to be careful about making a prediction about the position of the Galaxy in the past because z is influenced by the velocity in the past. The past meaning the moment of emission.
If Figure 3 is more realistic (compared to figure 2) than the recession speed of NGC 4258 (at present) could be much less than the value of 537 km/sec obtained using v = c * z. The same is true for the recession speed of UGC 3789 of 3325 km/sec at present.
If Figure 4 is true, then the most distant Galaxies (highest values of z now), which currently are assumed to describe the situation just after the Big Bang, could depict a situation much later as currently assumed, making it much easier to understand that the Universe looks Uniform even at large distances.

For a more technical discussion which starts from the concept that the universe is homogeneous see here: Variable Hubble Constant H . The conclusion is the same: Further away galaxies which large values of z are much closer.

The problem with the first Hubble's Law are:

1. At short distance until 7 Mpc (NGC 4258) the Hubble constant H is difficult to calculate because the redshift (the speed v1) of the Galaxy at moment of emission relative towards the Observer is large compared to the redshift caused by space expansion.
2. The measurement of the absolute distance using parallax of a galaxy like UGC 3789 at 46 MPC is difficult because of the small angles involved.
3. The validity of the assumption that at much larger distance (from point of emission) redshift increases linear with distance. Maybe locally (at present) there is much less space expansion than at further distances (in the past).

### Literature

1. A RELATION BETWEEN DISTANCE AND RADIAL VELOCITY AMONG EXTRA-GALACTIC NEBULAE By Edwin Hubble. Original document of 17 January 1929
2. Ned Wright's Cosmology Tutorial, FAQ, part 1 This document explains Hubble's Law
3. Hubble's diagram and cosmic expansion Robert P. Kirshner, Historical overview
4. Hubble's diagram and cosmic expansion Robert P. Kirshner, Historical overview
5. Hubble's Law
6. Misconceptions about the Big Bang Scientific American. In this document we read:
In expanding space, recession velocity keeps increasing with distance. Beyond a certain distance, known as the Hubble distance, it exceeds the speed of light. This is not a violation of relativity, because recession velocity is caused not by motion through space but by the expansion of space.
The reader should ask him or herself what the physical difference is. Space in principle is empty space, so what is it.
7. Steady State versus Big Bang Cosmology Chapter 1 Book
8. Alternative Cosmologies
9. A Redetermination of the Hubble Constant with the Hubble Space Telescope from a Differential Distance Ladder 5 May 2009. See page 53 figure 12 for the linear relation of cz and magnitude (luminosity distance)
We think the safest choice is to begin the measurement of the Hubble flow at z > 0.023 to avoid the uncertainty of the Bubble or other coherent large-scale flows.
10. A Revised Cepheid Distance to NGC 4258 and a Test of the Distance Scale 23 January 2001
11. A Comparison of Distance Measurements to NGC 4258 By Bruce Rout, 29 October 2009.
This document shows that NGC 4258 is at a distance of 7.16 Mpc.
12. MICRO-ARCSEC ASTROMETRY OF THE MILKY WAY AND BEYOND By M. J. Reid, 2008
At page 58 of this document you can read:
the recessional velocity of NGC 4258 is small (475 km per sec) and the likelihood of non-Hubble flow motions of approx. 200 km per sec precludes a direct estimate of H0.
13. The Megamaser Cosmology Project: I. VLBI observations of UGC 3789 M. J. Reid, 21 November 2008. This document shows at page 2/3 that NGC 4258 has a speed of 475 km/sec and at page 11 that UGC 3789 has a speed of 3325 km/sec.
14. Precision cosmology with H2O megamasers: progress in measuring distances to galaxies in the Hubble flow By J. Braatz. This document gives detail information About NGC 6323. See page 401
15. The Redshift-Distance and Velocity-Distance Laws By Edward Harrison. 28 July 1992
16. Determination of the Hubble constant By Wendy L. Freedman and Long Long Feng. September 1999
The current limit for detection of Cepheids with HST is a distance of about 30 Mpc . At these distances peculiar motions still can contribute 10–20% of the observed velocity.

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Created: 27 November 2009
Updated 4 December 2009
Updated 9 January 2010
Updated 20 Februari 2016