A Cosmological basis for E=mc^2

TL;DR

This paper explores the cosmological significance of the gravitational horizon R_h=ct, proposing it as fundamental to physics by linking it to the origin of rest-mass energy through gravitational binding energy.

Contribution

It introduces a novel perspective connecting the gravitational horizon R_h=ct with the origin of rest-mass energy via gravitational binding energy calculations.

Findings

The gravitational binding energy of a mass at the horizon equals mc^2.

Particles near R_h have energy transitioning from rest to escape energy.

The R_h=ct cosmology aligns well with observational tests.

Abstract

The Universe has a gravitational horizon with a radius R_h=c/H coincident with that of the Hubble sphere. This surface separates null geodesics approaching us from those receding, and as free-falling observers within the Friedmann-Lemaitre-Robertson-Walker spacetime, we see it retreating at proper speed c, giving rise to the eponymously named cosmological model R_h=ct. As of today, this cosmology has passed over 25 observational tests, often better than LCDM. The gravitational/Hubble radius R_h therefore appears to be highly relevant to cosmological theory, and in this paper we begin to explore its impact on fundamental physics. We calculate the binding energy of a mass m within the horizon and demonstrate that it is equal to mc^2. This energy is stored when the particle is at rest near the observer, transitioning to a purely kinetic form equal to the particle's escape energy when it approaches R_h. In other words, a particle's gravitational coupling to that portion of the Universe with which it is causally connected appears to be the origin of rest-mass energy.

Equations

$$d{s}^2=c^{2}d{t}^2-a^2(t)\left[\frac{d{r}^2}{1-k{r}^2}+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)\right].$$

$$a(t)=e^{f(t)},$$

$$r=R{e}^{-f},$$

$$dr=e^{-f}dR-\dot{f}rdt.$$

$$d{s}^2=c^{2}d{t}^2\left[1-{\left(\frac{R\dot{f}}{c}\right)}^2\right]+2\left(\frac{R\dot{f}}{c}\right)cdtdR-d{R}^2-R^{2}d{\Omega }^2,$$

$$d{\Omega }^2\equiv d{\theta }^2+{\sin }^2\theta d{\varphi }^2.$$

$$\Phi \equiv 1-{\left(\frac{R}{R_{\mathrm{h}}}\right)}^2,$$

$$c^{2}d{t}^2-a^{2}d{r}^2$$

$$d{s}^2=\Phi {\left[1+\left(\frac{R}{R_{\mathrm{h}}}\right){\Phi }^{-1}\frac{\dot{R}}{c}\right]}^2{c}^{2}d{t}^2-{\Phi }^{-1}d{R}^2-R^{2}d{\Omega }^2$$

$$p^{\mu }\equiv (E/c,p^R,p^{\theta },p^{\varphi }),$$

$$\Phi {\left[1+\left(\frac{R}{R_{\mathrm{h}}}\right){\Phi }^{-1}\frac{\dot{R}}{c}\right]}^2{\left(\frac{E}{c}\right)}^2-{\Phi }^{-1}{\left(m\dot{R}\right)}^2=K^2,$$

$$p^R=m\dot{R},$$

$$E^2=\frac{(cK{)}^2\Phi +(mc{)}^2{\dot{R}}^2}{{\left[\Phi +\left(\frac{R}{R_{\mathrm{h}}}\right)\frac{\dot{R}}{c}\right]}^2},$$

$$E(R_{\mathrm{h}})=m{c}^2.$$

$$m\dot{R}=mc\left(\frac{R}{R_{\mathrm{h}}}\right),$$

$$E^2=(m{c}^2{)}^2\left[1-{\left(\frac{R}{R_{\mathrm{h}}}\right)}^2\right]{\left(\frac{K}{mc}\right)}^2+(m{c}^2{)}^2{\left(\frac{R}{R_{\mathrm{h}}}\right)}^2.$$

$$E=m{c}^2$$

$$U^{\mu }=(c,\dot{R},0,0).$$

$$\frac{d}{dt}\left(p^{\mu }{p}_{\mu }\right)=m^2\frac{d{U}^{\mu }}{dt}{U}_{\mu }+m^2{U}^{\mu }\frac{d{U}_{\mu }}{dt}.$$

$$U_{\mu }\equiv g_{\mu \nu }{U}^{\nu },$$

$$g_{tt}\equiv \Phi {\left[1+\left(\frac{R}{R_{\mathrm{h}}}\right){\Phi }^{-1}\frac{\dot{R}}{c}\right]}^2$$

$$g_{RR}\equiv -{\Phi }^{-1}$$

$$\frac{d}{dt}\left(p^{\mu }{p}_{\mu }\right)$$

$$\frac{d}{dt}\Phi =\frac{d}{dt}{\Phi }^{-1}=0,$$

$$\frac{d}{dt}{\left[1+\left(\frac{R}{R_{\mathrm{h}}}\right){\Phi }^{-1}\frac{\dot{R}}{c}\right]}^2=2{\Phi }^{-2}\frac{R}{R_{\mathrm{h}}}\frac{\ddot{R}}{c}.$$

$$\frac{d}{dt}\left(p^{\mu }{p}_{\mu }\right)$$

Full text

1 Introduction

The Universe has a gravitational horizon with radius $R_{\rm h}=c/H$, where $H$ is the Hubble constant, coincident with the better known Hubble sphere [1, 3, 4, 5, 2]. Unlike its counterpart in the Schwarzschild and Kerr metrics, however, $R_{\rm h}$ is time-dependent so this surface may or may not eventually turn into an event horizon in the asymptotic future depending on the cosmic fluid’s equation of state. The gravitational horizon was formally introduced in ref. [1], though an unidentified predecessor appeared almost a century ago in de Sitter’s [6] own account of his now famous solution. In the intervening years, the choice of coordinates for which $R_{\rm h}$ appears explicitly in the metric was lost following the popularization of the comoving frame, principally by Friedmann [7]. In this paper, we shall have occasion to use the Friedmann-Lemaître-Robertson-Walker (FLRW) metric written in terms of both sets of coordinates.

The role played by $R_{\rm h}$ in any interpretation of the data is so important that a cosmological model based on its properties, known as the $R_{\rm h}=ct$ universe [9, 1, 3, 4, 2, 10], has already passed over 20 observational tests, typically better than $\Lambda$CDM. A summary of the model comparisons may be found in Table 1 of ref. [8]. An example of the impact $R_{\rm h}$ can have on our understanding of cosmological features is the role it played in resolving the question concerning whether or not cosmological redshift represents a new kind of time dilation, separate from the more conventional gravitational and Doppler effects. The answer is no—cosmological redshift is simply the product of these two [11], better known as the ‘lapse’ function in other applications of general relativity.

The concept of a gravitational radius in cosmology is not always easy to grasp because the observational evidence suggests the Universe is infinite. We are embedded within it, however, and the gravitational influence between us and another spacetime point depends solely on the intervening energy content. This may be understood quite easily in the context of the Birkhoff theorem [12] and its corollary (see also refs. [13, 1]). As such, every observer or particle—no matter where they are—is surrounded by a gravitational horizon a proper distance $R_{\rm h}$ away because the rest of the Universe exterior to this surface has a vanishing gravitational influence on the interior.

Such a limitation to our causal connectedness suggests a possible impact on fundamental physics. In this paper, we begin to examine this issue by asking a very basic—yet profound—question concerning the nature of rest-mass energy—specifically, whether it may be related in some way to a particle’s binding energy within the gravitational horizon.

We should emphasize at the outset that we are here making a clear distinction between the origin of inertia, i.e., rest mass, $m$, and the nature of rest-mass energy, $mc^{2}$. As far as we know today, the Higgs mechanism, with its SU(2) internal symmetry group, endows inertia to elementary particles that couple to the Higgs field [14, 15]. Why inertia is associated with an energy $mc^{2}$ is a different question.

2 The Friedmann-Lemaître-Robertson-Walker Metric

We begin with the FLRW metric for a spatially homogeneous and isotropic three-dimensional space, scaled by the expansion factor $a(t)$:

[TABLE]

The comoving coordinates used in this expession include the cosmic time $t$, an appropriately scaled radial coordinate $r$, and angular coordinates $\theta$ and $\phi$. The geometric factor $k$ is $+1$ for a closed universe, [math] for a flat universe, and $-1$ for an open universe. The high-precision measurements available today [16, 17] suggest that the Universe is flat, so we will assume the value $k=0$ throughout this paper.

As we proceed through this discussion, we shall see that $(ct,r,\theta,\phi)$ are the coordinates of a free-falling observer, analogous to a counterpart in the Schwarzschild or Kerr spacetimes. But for the latter, it has also been very useful to recast the metric in a form relevant to an accelerated observer—one who is at rest with respect to the central mass—and we shall similarly follow this procedure in the cosmological context. To do this, we introduce the proper radius, $R(t)\equiv a(t)r$, often used to express the distance that changes along with the expansion of the Universe. This proper distance $R(t)$ is a direct consequence of Weyl’s postulate applied to an isotropic universe [18], i.e., that no two worldlines in a cosmology satisfying the Cosmological principle should ever cross following the big bang—other than from local peculiar motions—which requires every distance in FLRW to be expressible as the product of an unchanging comoving length $r$ and a universal, position-independent function of time, $a(t)$.

We shall follow the procedure introduced in refs. [2, 5] to rewrite the FLRW metric in terms of $R(t)$. Writing the expansion factor in the form

[TABLE]

we put

[TABLE]

so that

[TABLE]

The metric in Equation (1) thereby becomes

[TABLE]

where, for convenience, we have defined

[TABLE]

Now introducing the function

[TABLE]

which will signal the dependence of the metric coefficients $g_{tt}$ and $g_{RR}$ on the proximity of $R(t)$ to the gravitational radius $R_{\rm h}$, the first two terms in Equation (5) may be rewritten as follows:

[TABLE]

We now consider the line element along the worldlines of particular observers, those that have $t$ as their proper time from one location to the next, i.e., comoving observers, as it turns out. Introducing the proper speed $\dot{R}\equiv dR/dt$ along these worldlines, we may then complete the square in Equation (8) and write Equation (5) as

[TABLE]

Some may see a similarity between this form of the metric and that used to derive the Oppenheimer-Volkoff equations for the interior of a star [19, 20] except, of course, that the latter is static, whereas both $R(t)$ and $R_{\rm h}(t)$ are functions of $t$ in FLRW.

3 Binding Energy

Let us now define the 4-momentum of a particle

[TABLE]

where $E$ is its energy, and $p^{i}$ are the usual spatial components, and consider the invariant contraction $p^{\mu}p_{\mu}$. For the metric coefficients in Equation (9), one has

[TABLE]

where $K$ is a constant (i.e., a scalar) yet to be determined, and we have assumed purely radial motion with $p^{\theta}=p^{\phi}=0$ and

[TABLE]

in terms of the particle’s rest mass $m$. Note that no additional factor, such as a time dilation, appears in Equation 12 because the cosmic time $t$, used in the derivative, is also the local proper time at every spacetime point in the cosmic fluid. In the Appendix, we demonstrate that the contraction of $p^{\mu}$ with itself, based on the definitions in Equations (10) and (12), is a scalar and a constant in the spacetime described by Equation (9). Equation (11) thus expresses the particle’s energy $E$ in terms of its momentum $m\dot{R}$ everywhere in the medium, starting from the observer’s location at the origin ($R=0$) all the way to the gravitational horizon at $R_{\rm h}$.

Let us re-write it in a somewhat more conventional form,

[TABLE]

and first consider what happens at the horizon. There $R=R_{\rm h}$ and $\dot{R}=c$, while $\Phi=0$. Clearly,

[TABLE]

But notice that this value comes—not from $K$, which one would naively have assumed ab initio—but rather from the momentum transitioning to its relativistic limit, i.e., $p^{R}\rightarrow mc$, while the contribution from $K$ itself gets redshifted away completely as a result of $\Phi\rightarrow 0$ when $R\rightarrow R_{\rm h}$. This result is quite remarkable because it tells us that the particle’s escape energy as it nears the gravitational horizon is what we would normally call its rest-mass energy $mc^{2}$. The emphasis here is on the phrase ‘escape energy’ because this value of $E$ is entirely due to $p^{R}$ at $R_{\rm h}$.

Assuming that the particle has no peculiar velocity at $R<R_{\rm h}$, we may also write

[TABLE]

and therefore the general expression for the total energy is

[TABLE]

A quick inspection of Equation (9) shows that in the $R_{\rm h}=ct$ universe, the metric coefficients $g_{tt}$ and $g_{RR}$ are time-independent. This is because both $R(t)$ and $R_{\rm h}$ are proportional to $t$. And as is well known in general relativity [13], energy is conserved along a particle geodesic—here represented by Equation (15)—when the spacetime metric is independent of time [21]. In addition, the fact that the $R_{\rm h}=ct$ universe has zero active mass, i.e., $\rho+3p=0$ [3, 4], means that the particle experiences zero net acceleration, so it cannot gain or lose energy from the background, and therefore $E$ in Equation (16) must be constant within the framework of $R_{\rm h}=ct$. But according to this energy conservation equation, $E$ can be constant only for one particular value of $K$, and that is $K=mc$, in which case

[TABLE]

everywhere and at all times.

This equally remarkable result tells us that the total energy $E$ can remain constant even though $p^{R}$ increases from [math] at the origin to its maximum value $mc$ at $R_{\rm h}$. We interpret this to mean that the particle’s binding energy $mc^{2}$ at the origin is gradually converted into kinetic energy as its proper distance from us nears our gravitational horizon, and $E$ becomes entirely kinetic when $R=R_{\rm h}$, but always equal to $mc^{2}$.

Notice also that we began our comparison of the gravitational horizon in cosmology with its counterpart in Schwarzschild and Kerr by emphasizing the fact that $R_{\rm h}$ changes with time. Yet none of the results, particularly Equations (14) and (17), are affected by this. Even as $R_{\rm h}$ increases with time, $E$ always remains constant and $p^{R}$ depends only on the ratio $R/R_{\rm h}$. So the value $E=mc^{2}$ and its transition from binding to kinetic energy (via Eqs. 13 and 15) remain valid forever. As long as a proton’s mass has remained constant in time, its rest-mass energy today is identical to its rest-mass energy minutes after the big bang.

4 Discussion

The quantity $E=mc^{2}$ may be interpreted as a gravitational binding energy because, according to the observer at the origin, this is how much energy the particle would need to free itself from its gravitational coupling to the Universe within $R_{\rm h}$. The region exterior to $R_{\rm h}$ does not participate in this gravitational interaction. It is apparently this $E$ that is gradually converted into kinetic energy (in the form of $p^{R}$), reaching its “escape” value $p^{R}c=(mc)c$ at the gravitational radius $R_{\rm h}$. Note that in this sense, $mc^{2}$ is literally the binding energy required to climb out of the gravitational potential well.

Mathematical consistency with these ideas is ensured by the invariance of the contracted 4-momentum vector, $p^{\mu}p_{\mu}$, which tells us exactly how the energy is changing in terms of the particle’s momentum. The physical descriptions we provide here inform our understanding of what is happening, but ultimately it is the invariance of the scalar $K$ that yields the dependence of $p^{R}$ on $R$. We do not actually have to calculate $E$ from the gravitational interaction itself. This is already done for us through the presence of $\Phi(R)$ in the metric. In other words, the redshift effect represented by $\Phi$ accounts for the gravitationl attraction the particle feels to the rest of the Universe within $R_{\rm h}$.

A more subtle point has to do with why the particle’s inertial mass is proportional (or even equal) to its gravitational mass. We do not attempt to broach this subject here, but as is well known, this is the basis for the Principle of Equivalence in general relativity. With it, we may use the particle’s inertia to characterize the strength of its gravitational interaction with the surrounding medium, so it is legitimate for us to ask what its gravitational binding energy is in terms of $m$. Of course, this is the reason we can interpret $mc^{2}$ as a gravitational binding energy in the first place. If inertia were unrelated to the gravitational mass, then there would be no physical reason at all for us to argue that the rest energy associated with $m$ should have anything to do with gravity.

When discussing such concepts, it clearly matters who the observer is. From the perspective of an observer fixed at the origin of the coordinates $(ct,R,\theta,\phi)$, the Universe is not static. Every particle moves away from him at the Hubble speed, $\dot{R}$, which increases steadily and reaches $c$ when $R=R_{\rm h}$. From his perspective, the cosmic fluid has a total energy commensurate with its momentum $p^{R}$. Thus, if the origin of a particle’s rest energy $mc^{2}$ were independent of its recessional velocity, the Hubble flow would be progressively more energetic as $R\rightarrow R_{\rm h}$ which, as we have seen, is not confirmed by the invariance of $p^{\mu}p_{\mu}$. So for this particular observer, the quantity $mc^{2}$ represents a blend of stored and kinetic energy, which transitions to $p^{R}c$ completely at the gravitational horizon.

When viewed in the comoving frame, however, the cosmic fluid is always at rest (other than for peculiar velocities that do not contribute to the true Hubble flow). Observers in this frame therefore see only the energy $E=mc^{2}$ corresponding to $p^{r}=0$. Strictly speaking, there is a different free-falling frame at each new location, so the particle’s rest energy is measured by different observers at different spacetime points. It is this switching from one observer to the next that replaces the variation of $p^{R}$ with distance in the accelerated frame.

Finally, it may be worth mentioning that the approach we have followed here in deriving our result has some overlap with the method commonly used to infer the mass-energy of so-called cosmological black holes. Unlike static black holes in a flat spacetime background, real black holes must necessarily be embedded within an expanding FLRW metric (see, e.g., refs [22, 23, 24, 25, 26, 27]). Modeling these extended bodies in a curved background introduces various degrees of coupling between their mass-energy and the geometry of the Universe at large, notably its apparent (or gravitational) horizon [28]. This in turn affects their dynamics and their own horizon. While this topic does not directly refer to the nature of rest-mass energy per se, the relationship between the enclosed energy of cosmological black holes and the type of background metric arises from the same gravitational interaction within a causally connected region that we have invoked to calculate the binding energy of a fundamental particle within the Universe’s horizon. Some issues revolving around how to best define masses and energy for cosmological black holes still remain unresolved, but the steps taken to couple the Kerr (or Schwarzschild) and FLRW metrics are based on similar physical principles that we have used in this paper.

5 Conclusion

The identification of rest-mass energy with the binding energy inside our gravitational horizon is thus quite compelling. Indeed, our argument is based entirely on core principles in general relativity. Were rest-mass energy due to something else, one would need to explain—within the framework of this theory—why the total energy at $R_{\rm h}$ is not greater than $mc^{2}$, in spite of the fact that $p^{R}\rightarrow mc$.

The success of the $R_{\rm h}=ct$ cosmology in providing such an elegant, accessible explanation for the origin of rest-mass energy adds to its credentials as a viable description of nature. Its principal divergence from $\Lambda$CDM is that it does not have a horizon problem, so it does not have or need inflation to account for the uniformity of the microwave background across the sky [29]. Without inflation [30, 31], the standard model could not survive, yet even after four decades of study, we still do not have a complete, self-consistent understanding of the inflaton field (see, e.g., refs. [32, 33]). Perhaps this too is an indication that inflation never happend, pointing to the $R_{\rm h}=ct$ universe as the only viable cosmology. Additional high-precision tests are underway [34], and we may have a definitive answer within a matter of years.

6 Appendix

In this appendix, we demonstrate that the contraction $p^{\mu}p_{\mu}$, with the four-momentum defined in Equations (10) and (12), is a scalar and a constant in the spacetime given by the metric in Equation (9). In doing so, we recall the discussion concerning the selected worldlines with proper speed $\dot{R}\equiv dR/dt$ preceding this equation, and we simplify the procedure by invoking the condition of zero peculiar motion everywhere, i.e., $\dot{r}=0$. As such, $\dot{R}=\dot{a}r=HR$, where $H$ is the Hubble constant $H\equiv\dot{a}/a$. In addition, it is trivial to see that $\dot{R_{\rm h}}=c$ in the $R_{\rm h}=ct$ universe, since $R_{\rm h}\equiv c/H$ (see, e.g., refs. [2, 10]). Since the Universe is isotropic and homogeneous, the geodesics are radial so, with zero peculiar velocities, we may also write the four-velocity $U^{\mu}\equiv dX^{\mu}/d\tau=dX^{\mu}/dt$, where $X^{\mu}=(ct,R,\theta,\phi)$, as

[TABLE]

Let us now consider the time evolution of $p^{\mu}p_{\mu}$, with $p^{\mu}\equiv mU^{\mu}$. That is, we shall proceed to evaluate the derivative

[TABLE]

With the four-velocity in Equation (18), and its covariant analogue

[TABLE]

in which only the metric coefficients

[TABLE]

and

[TABLE]

are non-zero, Equation (19) becomes

[TABLE]

In evaluating the right-hand side of this equation, it will be helpful to see that

[TABLE]

and that

[TABLE]

Therefore,

[TABLE]

so the contraction of the four-momentum $p^{\mu}$ is clearly a scalar and a constant in this spacetime.

\ack

I am grateful to the Instituto de Astrofísica de Canarias in Tenerife and to Purple Mountain Observatory in Nanjing, China for their hospitality while part of this work was carried out. I also acknowledge partial support from the Chinese Academy of Sciences Visiting Professorships for Senior International Scientists under grant 2012T1J0011, and from the Chinese State Administration of Foreign Experts Affairs under grant GDJ20120491013.