The 1905 Einstein equation in a general mathematical analysis model of Quasars

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1 DePaul University From the SelectedWorks of Byron E. Bell Spring May 3, 2010 The 1905 Einstein equation in a general mathematical analysis model of Quasars Byron E. Bell Available at:

2 Byron E. Bell DePaul University and Columbia College (CC) Chicago The 1905 Einstein equation in a general mathematical analysis model of Quasars bbell16@depaul.edu, bbell@colum.edu Abstract The 1905 wave equation of Albert Einstein is a model that can be used in many areas, such as physics, applied mathematics, statistics, quantum chaos and financial mathematics, etc. I will give a proof from the equation of A. Einstein s paper Zur Elektrodynamik bewegter Körper it will be done by removing the variable time (t) and the constant (c) the speed of light from the above equation and look at the factors that affect the model in a real analysis framework. Testing the model with SDSS-DR5 Quasar Catalog (Schneider +, 2007). Keywords: direction cosine, apparent magnitudes of optical light; ultraviolet (m u ), green (m g ), red (m r ), more red (m i ), and even more red (m z ), redshift (z), Regression equation. Introduction The work of Albert Einstein from the 1905 on the Maxwell-Hertz equations can be used in at least two dependent variables redshift (z) and Right Ascension (R.A.), coefficient of the variables m ugriz of the five independent variables cosine directions ( T ugriz ). Using the facts from Einstein 1905 the paper Zur Elektrodynamik bewegter Körper and the section Theorie der Dopper's Prinzip und Aberration. Taking the original equations from the above work will show a strong relationship between the independent variables of direction cosine of the T u,t g, T r, T i, T z and the dependent variables z and R.A.. A proof will be given of the equations without the variable time (t) and the constant (c) the speed of light. And data will show a high correlation between the independent variable(s) and the dependent variable(s) of the model(s). And there is also a strong relationship between the apparent magnitudes of optical light; ultraviolet (m u ), green (m g ), red (m r ), more red (m i ), and even more red (m z ). The software packetage that was used in this study is SPSS 11.5, with the data of SDSS-DR5 Quasar Catalog (Schneider +, 2007).

3 Proposition A proof from the equation of Albert Einstein s paper Zur Elektrodynamik bewegter Körper will be done by removing the variable time (t) and the constant c the speed of light from the equation below on the left hand side (LHS) to become what is on the right hand side (RHS): =f(t u,t g, T r, T i, T z )=t ) = f(t u,t g, T r, T i, T z ) Where the apparent magnitudes of optical light, m ugriz : [ultraviolet (u), green (g), red (r), more red (i), even more red (z)], ugriz of quasars then T u, T g, T r, T i, T z is direction cosine of the apparent magnitudes (m ugriz ). Proof: Given =f (T u,t g, T r, T i, T z ): =f(t u,t g, T r, T i, T z ) =t ) (1) Then set (RHS) of (1) the variable t to zero (t =0) and then the equation becomes the following: = ) (2) Then set (LHS) of (2) to zero then it becomes the following: 0= ) (3) Then multiply both sides of (3) by c and the above equation becomes following f(t u,t g, T r, T i, T z ) = (4) Then finally equation (4) becomes equation (5) =f(t u,t g, T r, T i, T z ) = (5) QED

4 Analysis of Redshift (z) In understanding the relationship between redshift (z) and apparent magnitudes (m ugriz ) in the work of (Schneider +, 2007) is a must to examine when studying Quasars. The using of linear regression and the study of black holes by (Ferrarese, Merritt 2000) is a groundbreaking work in astrostatistics and astro-physics. The linear regression of redshift (z) and cosine directions ( T ugriz ) as input variables is used in this current work. Regression Variables Entered/Removed b,c Model Variables Entered Variables Method 1 T u,t g, T z Enter a. Tolerance= Limits reached. b. Dependent Variables: z c. Linear Regression through the Origin Model Summary Model R R Square Adjusted R Square Std. Error of the Estimate a. For regression through the origin (the no-intercept model), R Square measures the proportion of the variability in the dependent variable about the origin explained by regression. This CANNOT be compared R Square for models which include an intercept. b. Predictors: T u,t g, T z Model 1 Regression Residual Total ANOVA c,d Sum of Squares df Mean Square F Sig a b a. Predictors: T u,t g, T z b.this total sum of squares is not correct for the constant because the constant is zero for regression through the origin. c. Dependent Variable: z d. Linear Regression through the Origin - 3 -

5 Coefficients a, b Model t Sig. 1 T u T g T z a. Dependent Variable: z b. Linear Regression through the Origin In the Model Summary Table is R=.930, R-Square=.864, Adjusted R Square =.864 The equation of Coefficients Table is =f(t u,t g, T z ) = + (s u ) (s g ) (s z ) Hypothesis F-test H 0: ρ=0 H 1 :ρ 0 F= , Sig. of T z, T u, T g = Alpha >Sig. Reject H 0 Alpha < Sig. Accept H 0.05> Reject H 0 The decision is to reject H 0 at the Alpha=.05 level. Therefore it is a statistically significant relationship between the dependent variable redshift (z) and the independent variables of the cosine directions (T u,t g, T z )

6 Analysis of Right Ascension (R.A.) Right Ascension (R.A.) is the longitude of an object in space and the relationship to cosine directions is very important in understanding Quasars in apparent magnitudes terms. The above model used multiple linear regression also called ordinary leastsquares (OLS) of z on T z, T u,t g. Then in this statistical analysis, ordinary least-squares (OLS) of R.A. also as above the cosine directions (T z, T u,t g ) are good predictors of the Right Ascension (R.A.). Method of using this process and more is in (Isobe, Feigelson, Akritas, Babu 1990) and (Feigelson, Babu 1992). Regression Variables Entered/Removed b,c Model Variables Entered Variables Method 2 T u, T g, T z Enter a. Tolerance= Limits reached. b. Dependent Variables: R.A. c. Linear Regression through the Origin Model Summary Model R R Square Adjusted R Square Std. Error of the Estimate a. For regression through the origin (the no-intercept model), R Square measures the proportion of the variability in the dependent variable about the origin explained by regression. This CANNOT be compared R Square for models which include an intercept. b. Predictors: T u, T g, T z Model 2 Regression Residual Total ANOVA c,d Sum of Squares df Mean Square F Sig. 2.38E a 4.46E E a. Predictors: T u, T g, T z b.this total sum of squares is not correct for the constant because the constant is zero for regression through the origin. c. Dependent Variable: R.A. d. Linear Regression through the Origin - 5 -

7 Coefficients a, b Model t Sig. 2 T u T g T z a. Dependent Variable: R.A. b. Linear Regression through the Origin In the Model Summary Table is.040 R=.918, R-Square=.842, Adjusted R Square =.842 The equation of Coefficients Table is =f(t u,t g, T z ) = + (s u ) (s g ) (s z ) Hypothesis F-test H 0: ρ=0 H 1 :ρ 0 F= , Sig. of T z, T u, T g = Alpha >Sig. Reject H 0 Alpha < Sig. Accept H 0.05> Reject H 0 The decision is to reject H 0 at the Alpha=.05 level. Therefore it is a statistically significant relationship between the dependent variable Right Ascension (R.A.). and the independent variables of the cosine directions (T u,t g, T z )

8 Analysis of Even more red light (m z ) The work of (Bell 2008) the author studied variability of errors in apparent magnitudes of the SDSS data set in a Autoregressive Conditional Heteroskedasticity (ARCH) method from econometrics. Regression Variables Entered/Removed b,c Model Variables Entered Variables Method 3 T u, T g, T i Enter a. Tolerance= Limits reached. b. Dependent Variables: m z c. Linear Regression through the Origin Model Summary Model R R Square Adjusted R Square Std. Error of the Estimate a. For regression through the origin (the no-intercept model), R Square measures the proportion of the variability in the dependent variable about the origin explained by regression. This CANNOT be compared R Square for models which include an intercept. b. Predictors: T u, T g, T i Model 3 Regression Residual Total ANOVA c,d Sum of Squares df Mean Square F Sig E+07 a b a. Predictors: T u, T g, T i b.this total sum of squares is not correct for the constant because the constant is zero for regression through the origin. c. Dependent Variable: m z d. Linear Regression through the Origin - 7 -

9 Coefficients a, b Model t Sig. 3 T u T g T i a. Dependent Variables: m z b. Linear Regression through the Origin In the Model Summary Table is R=.999, R-Square=.998, Adjusted R Square =.998 The equation of Coefficients Table is m z = m u T u + m g T g + m i T i (s u ) (s g ) (s i ) Hypothesis F-test H 0: ρ=0 H 1 :ρ 0 F=1.6E+07, Sig. of T u, T g, T i = Alpha >Sig. Reject H 0 Alpha < Sig. Accept H 0.05> Reject H 0 The decision is to reject H 0 at the Alpha=.05 level. Therefore it is a statistically significant relationship between the dependent variable Even more red light (m z ) and the independent variables of the cosine directions (T u, T g, T i )

10 Summary There is a statistically significant relationship between the dependent variables of Redshift (z), Right Ascension (R.A.) and Even more red light (m z ) from the independent variables of direction cosine (T ugriz ). More studies are needed in this area of astrostatistics like bayesian statistics. References Bell, B. 2008, From Asteroids to Cosmology. International Symposium. Einstein, A. 1905, Annalen der Physik. 17:891. Feigelson, E., Babu, J ApJ. 397, 55. Ferrarese, L., Merritt, D. 2000, arxiv: astro-ph/ v1. Isobe, T., Feigelson, E., Akritas, M., Babu, J. 1990, ApJ. 364, 104. Schneider, D , A J 134,