## Hubble's Law

### 7 Questions:

 Question 1 What is Hubble's Law Question 2 Is there only one Hubble's Law ? Question 3 The Doppler shift law v = c * z Question 4 What is the definition of distance (velocity) ? Question 5 Is the Hubble constant constant in time ? Question 6 By observing z "now" can we say anything about the present position and velocity ? Question 7 What prove is there for Hubble's Law ?

### Answer Question 1 - Second Hubble's Law

Hubble's Law states that there is a linear relation between the speed v of a Galaxy and the distance of that Galaxy. This law is written as v = H * d. For more details See: Hubble's Law in Wikipedia

### Answer question 2 - First Hubble's Law

There actual are two Hubble's Law.
• There is a law which states that there is a linear relation between the measured redshift z of a Galaxy and the distance of that object. This law is written as: z = H/c * d. This is the first Hubble's Law
• The law explained in Q1: v = H * d is called the second Hubble's Law

### Answer Question 3 - v = c * z

The law v= c * z describes what is called the doppler shift. The doppler shift is a frequency shift transmitted by a moving light source. Suppose a light source at rest with the Oberver transmits light with frequency f1. Next suppose that this source has a speed v relativ to the Observer than this frequency changes from f1 to f2, or with "Delta f" = f2 - f1. Dividing "Delta f" by f1 gives the value of z. Multiplying z with c gives v, the speed of the light source.
For more detail see: v = c * z

The Doppler shift law v = c * z can be used to derive the second Hubble's Law from the first Hubble's Law:
Start from the first and multiply both sides with c than you get: z * c = H/c * d * c. Or: v = H * d

### Answer Question 4 - distance

In actual fact there are two distances involved: The past distance and the present or proper distance
• The past distance is the distance at moment of emission of the ligt signal. That was the distance of the Galaxy, which we see now.
• The present distance is the distance at the moment of observation i.e. the moment that the light signal reaches the Observer
The same can be said for velocity: past velocity and present velocity.
• The past velocity is the velocity of the Galaxy at the moment
• The present velocity is the velocity at the moment of observation.

The issue is if we can claim based on local observations anything about global situations.
If we observe a Supernovae then we now that an explosion has happened in the past. Maybe certain observations allow us to calculate when, where and which the speed was at the time of this explosion. The question is do those same observations allow us calculate the present position and speed of what is left over of this exploded star? Read more in the section about prove.

### Answer Question 5 - H constant

In fact there are two issues: local versus global and constant in time.
• Accordingly to current understanding space expansion is homogeneous. That means there exists a linear relation between the present expansion velocity versus the present distance.
• However this does not imply that the expansion velocity at a fixed distance is and was always the same. In fact inflation theory challenges the idea that this is a constant, because the inflation theory claims that space expansion initially, just after the Big Bang, was much larger than presently.

### Answer Question 6 - relation present position and velocity

IMO the z we measure now is a function of what happened between the moment that the lightflash (supernovae) was emitted and the state of the space that this flash traversed during the time that the lightflash travelled from its source towards its destination i.e. the present.
IMO as such by observing z now any claim about the present position and velocity is rather speculative.
IMO it is also speculative to claim anything about the past position and velocity at the moment of emission.
Both answers are not in conflict with the idea that space expansion at any given moment is homogeneous.

In the program Bigbang3.xls in Excel z is simulated for 3 different expansion scenario or world views
For a description and a copy select: Excel Program: Bigbang3.XLS What the program demonstrates is that the value of z we measure now, using the same situation at the moment of emission (initial conditions), is different for each expansion scenario.

### Answer Question 7 - prove

The prove of Hubble's consists of two parts:
• The proof that there exists a linear relation between z and distance. This is the first Hubble's Law
• The proof that there exists a linear relation between velocity and distance. This is the second Hubble's Law
The problem is when you read in the literature about Hubble's Law in most cases the subject is the second Law. In those article's the speed is mentioned as if that is a measured quantity. The problem is that is not the case. What is measured is z and the Doppler shift law v = c * z is used to calculate the speed.
The next step is to claim that both the speed and the distance represent proper (present) values.
In principle you can do that but if you do that you need independent measurements of speed and distance to back up this claim. IMO this is never done because it is very difficult.

To prove the first Hubble's law is easier but than you have to show what type of distances are involved: past distances or proper distances.
On the other hand at the scale of the Andromeda Galaxy First Hubble's law is not valid because that Galaxy approaches implying that you cannot calculate the z versus distance relation at that distance.
The next set of Galaxies to study in order to study the first Hubble' Law are: NGC 4258, UGC 3789 and NGC 6323. Unfortunate literature does not show independent measured distance values using parallax. Starting point are measured values of z and previous calculated values of H. Using the second Hubble's Law the distances are calculated. But that is not what we want, partly because it is not known what those distances represent. What we want is to show if there exists a linear relation between z and distance and than you should not use H.

### Prove versus a model

In the discussions about Hubble's Law the concept of models is used. Models are mathematical equations in order to describe the behaviour of systems. Example's of models are Newton's Law, SR and GR. Newton's Law can perfectly be used in order to calculate the present position of the planets of the Solar system and the present position of certain stars based on observations which represent the past.
In order to prove or to demonstrate Hubble's Law also models are used. A model in this case is for example the cosmological principle which claims that the Universe is homogenous and isotropic.
The second Hubble's law which claims that v = H * d using present values for velocity and distance fits perfectly in that scheme. The problem is that You first have to prove the cosmological principle using something else than the second Hubble's law.

### Reflection - the program "Hubble" and "VB Hubble"

For an critical evaluation of Hubble's Law read: There is also a simulation available which demonstrates the conflict between Hubble's Law and observation.
In this program (Hubble.bas) the simulation consists of a binary star system and a third star.
The conflict is that the third star can be ejected at almost any speed.
For all the details select this: Program "Hubble". Hubble's Law with 3 stars
The same program is also available in Visual Basic 5.0: "VB Hubble"
For all the details select: VB Hubble operation

### Feedback

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Original: 17 March 2010
Modified: 16 Februari 2016