For some reason, the Hubble tension remains a perplexing issue in cosmology. It involves a discrepancy between two methods of measuring the Hubble constant, which quantifies the rate of the universe's expansion. One method relies on measurements of the cosmic microwave background (CMB) radiation from the early universe, while the other utilizes observations of supernovae and other cosmic objects in the local universe. The Hubble constant values obtained from these two methods do not align, creating tension within the scientific community. Researchers are investigating whether this inconsistency points to "new physics" beyond our current understanding or if it can be resolved with improved measurements and methods, appearing to completely ignore the fact that indeed both methods may be serious flawed, necessitating an alternative explanation.

As with many of the apparent "mysteries" that have arisen in modern physics the Hubble tension is once more easily explained and yet again all that is required is high school physics. the solution to the problem revolves around a simple misinterpretation of velocity. Strangely the solution begins with an apparently unconnected equation, Schwarzschild's equation for escape velocity;

$${v_{escape}} = \sqrt{\frac{2 G m}{r}}$$

the equation itself can be refined as it is understood that mass equals the product of density and volume. Consequently, these values can be substituted into the Schwarzschild equation. Being that the volume of a sphere is \(\frac{4\pi r^3}{3}\) this can also be substituted into the Schwarzschild equation resulting in the following;

The result is an equation that looks exactly like the familiar Friedmann equation which is the very basis of the expanding universe theory.

$${v_e}^2 = \frac{8 \pi G r^2 \rho}{3}$$

The single difference however is subtle, inasmuch as the velocity in the Friedmann equation purportedly represents a "recession velocity" and not actually the mathematical fictional value of the escape velocity as in the Schwarzschild equation.

$${H}^2 = \frac{8 \pi G r^2 \rho}{3}$$

In actual fact the claim made by Friedmann that his equation which is identical in form and function to Schwarzschild's is derived from General Relativity thereby using a velocity claimed to be a velocity of recession rather than escape.

Escape velocity is the minimum speed an object needs to break free from the gravitational pull of a planet, moon, or other celestial body without any further propulsion. However, recession velocity refers to the speed at which galaxies and other celestial objects are moving away from each other due to the expansion of the universe. Mathematically proving the error is particularly simple, by substituting the kinetic energy and calculating the acceleration the resultant value is;

Imagine the following scenario containing a line of trees on each side of the road

These trees stretch off to the distance and it is known know far down the road the trees stretch, maybe a couple of miles or so. However what the physicists really want to know is how fast the trees are moving away. The answer is simple, any physicist can tell you, the velocity is the derivative of the postion with respect to time. When they want to know how much the trees are accelerating away, all that is needed is to differentiate the velocity function.

The issue with using this calculation becomes evident from the example; it’s a circular argument arising from an initial implicit assumption that the trees are accelerating away. Without this assumption, the calculation provides no meaningful conclusions. To understand this mathematically, see the following:

To calculate velocity using calculus, you need the position function \(x(t)\), which describes the position of an object as a function of time. The velocity \(v(t)\) is the derivative of the position function with respect to time, being expressed as:

$$v(t) = \frac{dx(t)}{dt}$$

Having the position as a function of time \(x(t)\), the velocity can be found by differentiating \(x(t)\) with respect to time \(t\). Once the velocity function \(v(t)\) is known the acceleration can then be found by differentiating the velocity function;

$$a(t) = \frac{dv(t)}{dt}$$

The question arises: is this actually the method being employed to determine the universe's acceleration, and are there any alternative ways to show that this is the case? Fortunately, the response is yes. From published articles the average acceleration is found to be approximately \(\text{70 m/s/Mpc}\). As such it can be asked what energy would be required to produce such an acceleration. After conversion to SI units and basic calculation it is found that the value needed is approxiamtely \(2.179\ × 10^{-18}\ \text{J}\), a value of energy required is so small that it would appear to be at a quantum level. Consequently, if a calculation is made using the escape velocity of the ground state electron in the hydrogen atom the resultant energy is;

$$2.179\ 872\ 3610 × 10^{-18}\ \text{J}$$

This of course is expected considering that the mass and energy of the universe is unknown and manipulation of the equation is so extensive, that nothing remains in the calculation representing actual physical values apart from the most basic units of measure.