SIMPLE EXPERIMENT TO VERIFY IF GRAVITATIONAL RED-SHIFTING IS TRUE

ABSTRACT

 This home experiment could prove if gravitational shift exists, and verify Δf/f = gh/c².

Even when simple, and using off-the-shelf elements, the use of a large amount of

seconds allows the possible test of red-shifting, if the difference in the pulses count

increases constantly and regularly every Tᴺ = c²/(f₁gh) sec = 40,000 sec, being f₁

the frequency of a simple MW cavity oscillator, tuned at 10.00 Ghz. The readings

of two 50 bit counters are periodically collected by a PC, which displays the value

of the counters and its difference. The experiment is based on the continuous

phase accumulation of a 10 Ghz signal and its receiving part, 22.2 m high, where

the gravitational effect should manifest as being Δf/f = - gh/c² = -2.49E-15.

 

 

FREQUENCIES AND DIFFERENCES IN COUNT OF PULSES.

 

Box 1 operates with a frequency f₁ = 10.00000 Ghz. The pulses to be counted 

continuously in N₁ are 0.1 nsec pulses.

 

Box 2 operates with a frequency  f₂, received from the 10.00000 Ghz transmitted from

the ground level, 22.2 m down.  f₂ =  f₁ + Δf = f₁ - 0.000025 Hz.

 

This is because Δf/f = - gh/c², if Einstein gravitational shift prediction is correct. 

 

Then, f₂ =  9,999,999,999.999975 Hz, and Δf/f₁ = (f₂ - f₁)/f₁ = -2.49E-15

 

If Einstein's gravitational shift prediction doesn't exists, f₂  =  f₁ = 10.000000 Ghz.

 

The difference |Δf₁| = 0.000025 Hz creates a difference ΔN = N₁-N₂ = +1 count,

accumulative with a period 1/|Δf₁| = 40,000 seconds. This means about +2 counts/day 

or +15/week.

 

SUBSYSTEMS  

 

Besides two DirecTV antennae, which are fed directly by the MW power amplifiers,

the following subsystems are used:

 

Box 1: Contains the 10.00000 Ghz cavity oscillator (f₁), the N₁ counter, the count 

memory, updated every 0.4295 seconds with timer T^G₁, the data count N₁ serializer

and the link with the PC, where the packetized reading of N₁. It also contains the 

DC power feed for the Box 2, and two outputs (USB ports O₁ and O₂) to allow the 

PC to receive the packets from both boxes. Each packet contains 64 bits of data

digital count plus a sequential number.

 

Box 2: Contains a 10.00000 Ghz pass band filter, a Ghz power amplifier, the N₂ 

counter, the count memory, updated with T^G₂ every 0.4295 seconds, the data 

count N₂ serializer and the link with the PC, to send the packetized count. 

The cable used for packetized data transfer to the PC, through Box 1, also contains

the DC power feed.

 

PC: Contains the SW to compute N₁ and N₂ every T^G = 0.4295 seconds.  This signal

alert PIC1 and PIC2 to transmit data to two separate USB ports of the PC. 

With N₁ and N₂ refreshed every 0.4295 sec, the PC SW can compute differences

indefinitely (hours, days, months).

 

SOFTWARE: Besides elementary binary arithmetic, to present ΔN = N₁ - N₂  data, the

software has to make corrections due to different delay sources in the analog domain:

 

1) 74 nsec for the signal propagation delay between Antenna 1 and Antenna 2.

2) 20 nsec (to be measured correctly) between the input of the N₁ counter and

the input of the N₂ counter, due to the MW filter, delays in the RF amplifiers, etc.

 

The total delay is constant, and has to be deducted from ΔN = N₁ - N₂, as it represents

a difference of about 940 pulses between both counters.

 

Counter reading times T^G₁  and T^G₂, and general timing T^G = 0.4295 sec

 

T^G₁  and T^G₂ are derived from the signals S₁ and S₂ by dividing each one by 2³².

The division creates low frequency gating signals, used for periodic reading of 

each counter N. The very small difference ΔTᴳ = Tᴳ₂ - Tᴳ₁ is in the order of one

femtosecond, which makes Tᴳ₁ = Tᴳ₂ = Tᴳ, to compute the time of readings.

 

f₁ = 10,000,000,000.0000000 Hz

T₁ = 0.1E-09 s
Tᴳ₁ = T₁ . 2³² = T₁ . 4,294,967,296 = 0.429496729600000 s


f₂ = f₁ + Δf = f₁ - 0.0000249 Hz = 9,999,999,999.9999751 Hz
T₂ = 1/(f₁ + Δf) = 1/f₁ (1 - 2.49E-15) s = T₁ (1 + 2.49E-15) s = 0.1E-09 s + 2.49E-25 s
Tᴳ₂ = T₂ . 2³² = T₂ . 4,294,967,296 = 0.429496729600000 s + 1.069446857E-15 s

ΔTᴳ = Tᴳ₂ - Tᴳ₁ = 1.069446857E-15 s (≈ 1 femtosecond)


For any practical purpose, Tᴳ₂ = Tᴳ₁, so Tᴳ = 0.429496729600000 s ≈ 0.4295 s is the

gating signal that is used to store the counts at Box 1 and Box 2, as the true real time clock.

EXAMPLE OF READINGS IN THE PC SCREEN

This is an example of some readings every random multiples of 20,000 T^G (8,590 sec):

If gravitational shift exist, then every Tᴺ = c²/(f₁gh) = 40,000 sec, ΔN increments in 1.

 

Time (s)          N1                                         N2                       ΔN

   0                      0                                          0                          0

8590            85900000000000           85900000000000           0

34360        343600000000000           343600000000000         0

42950       429500000000000           429499999999999         1

60130       601300000000000           601299999999999         1

77310       773100000000000           773099999999999         1

85900        859000000000000           858999999999998         2

103080    1030800000000000           1030799999999998       2

120260    1202600000000000           1202599999999997       3

154620    1546200000000000           1546199999999997       3

163210    1632100000000000           163299999999996         4

197570    1975700000000000           1975699999999996       4

206160     2061600000000000          2061599999999995       5

240520     2405200000000000          2405199999999995       5

249110      2491100000000000          249199999999994         6

257700      2577000000000000         2576999999999994       6

 

 THE PC SOFTWARE.

 

The experiment can prove if gravitational shift in frequency of a single 

EM wave exist, due to the simplicity of the method: counting accumulative

difference of pulses between two 50 bits counter.

 

As the PC receives, every 0.4295 seconds, the readings of  N₁ and N₂, it

can compute indefinitely the difference ΔN = N₁ - N₂, even if the counters

overflow and restart from zero.

 

Each packet received in the PC every T^G seconds  is time-stamped by the

PC RTC clock, which is accurate enough for the magnitudes of time involved.

The PC can store more than 400,000 readings per day indefinitely, which allows

computing at which value of  Δf₁/f₁ (- gh/c² or other), the count increments in one. 

 

The structure of the packet that the PC reads is:

[PIC Number] 

[Packet Number] 

[50 bit counter reading]

This packet is stored in the PC with his RTC (Real Time Clock) value at the moment 

of reception. So, what is stored in the memory, for further calculations, is the following 

data structure:

[PIC Number] 

[Packet Number]

[50 bit counter reading] 

[PIC Continuous Time] 

[PC RTC timestamp]

As both Packet Number word on each PIC is reset to zero under command of the 

operator of the system, the TRUE CLOCK (same for both Boxes) is embedded into

each packet prior to its transmission, and is THE ONE that has to be considered 

(for the purity of the experiment) when making averages, display data, etc.

This clocks (one per PIC) compute time as TP = 0.4295 x [Packet Number] seconds, 

and last 16,384 packets.

Is the duty of the PC, with his timestamp on each packet received, to increase the

[PIC Continuous Time] field by 7,037 seconds when TP restart from zero.

In this way, the "experiment time" can be computed forever, in units of 0.4295 seconds, 

showing this time in the field [PIC Continuous Time], which is stored along the rest of

the packet data in an ad-hoc database.

The only clock valid for the experiment is the one present in the field [PIC Continuous Time].

In this way, the PC can compute the elapsed time between integer increments of the 

difference between counters, with an error lower than Tᴳ/Tᴺ = 42.95/40,000 % = 0.01%.

And, as Tᴺ = c²/(f₁gh) = 1/Δf, if happens, "Einstein's right" can be measured with this error.

SIMPLE MATHEMATICS BEHIND THE EXPERIMENT

 

Einstein gravitational red-shifting of electromagnetic waves is expressed, for

low heights, as:

 

Δf₁/f₁ = (f₂ - f₁)/f₁ = - gh/c² 

f₂ =  f₁ - f₁gh/c² 

For f₁ = 10.000000 Ghz and h = 22.2 m, f₂ =  9,999,999,999.999975 Hz

The tiny difference of 0.000025 Hz can be measured digitally, only if enough time

is allowed to pass, so one integer increase of counts happens due to Δf.

 

With the frequency f₁ used, the time Tᴺ to produce a change ΔN = +1  is 

Tᴺ = 1/|Δf₁| = 1/|(f₂ - f₁)| = c²/(f₁gh) = 40,000 seconds. In one day, ΔN = +2 

and, in one week, ΔN = +15. The difference increases every Tᴺ seconds until

the experiment is stopped, or something breaks down.

  

If this happens, then the existence of the gravitational red-shifting can be

considered as true, and Einstein was right. If no counts are accumulated

in ΔN, except random glitches of +1, then the gravitational shift doesn't exist.

 

ANALOG COUNTERPART OF THE DIGITAL DIFFERENCE OF COUNTS

 

In the analog world, the signal at the bottom can be represented by:

 

S₁(t) = A₁ sinΦ₁(t) = A₁ sin(ω₁t)

 

The signal received at the top, after filtered and amplified, can be represented by

 

S₂(t) = A₂ sinΦ₂(t) = A₂ sin(ω₂t + θ) = A₂ sin[(ω₁ + Δω) t + θ] 

S₂(t) = A₂ sin[ω₁ (1 - gh/c²) t + θ]

 

The constant θ represents a constant number of radians, due to 

delays originated in:

 

1) The time invested for the signal S₁ to travel 22.2 m, which is 

0.0000000740 sec. This time comprehend 740 cycles of S₁, 

representing a constant delay of 4,649.56 radians.

 

2) The natural delay of electric signals propagating through the coax 

cables, the power amplifiers, the reception filter and other minor 

sources of delay. It's estimated as being a constant value of 20 nsec 

end-end. Traduced in radians, it represents 200 x 2π radians, or

1,256.64 radians.

 

θ = 4,649.56 rad + 1,256.64 rad = 5,906.2 rad

 

The phases Φ₁(t) and Φ₂(t) are a linear function of t, and increase 

continuously, without end, unless the source is turned off or 

something breaks down.

 

The phase difference between S₁(t) and S₂(t), therefore, increase 

constantly with time:

 

ΔΦ = Φ₂(t) - Φ₁(t) = ω₁t  - ω₁gh t/c² + θ - ω₁t  =  - ω₁gh t/c² + θ 

ΔΦ = - 0.000156451 t + 5,906.2

 

Taken in absolute value, |ΔΦ| = |- 0.000156451  t + 5,906.2|.

 

To make more clear the analogy with the digital word, |ΔΦ| has 

to be expressed in units of 2π radians, so the difference is now 

expressed as:

 

K(t) =|ΔΦ|/2π =  |-2.49E-05 t + 940| = |- t/40,160.7232 + 940|

 

It is evident  that the analog phase difference K(t) (discretized) 

increments in ONE CYCLE every 40,160.7232 seconds, 

indefinitely, as time passes.

 

But 40,160.7232 seconds is just 1/|Δf₁| = 1/|f₂ - f₁| = c²/(f₁gh), 

which contains the Einstein's expression for gravitational shift in 

the small height h.