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JamalS
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Gravitation couples to anything within the stress-energy tensor, as dictated by the field equations,

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi G T_{\mu\nu}$$

Charge and angular momentum both affect the curvature of spacetime, as they affect the metric. For example, consider a spinning charged black hole, desribed by the Kerr-Newman metric,

$$\mathrm{d}s^2 = -\left( {\mathrm{d}r^2 \over \Delta + \mathrm{d}\theta^2} \right)\rho^2 + \left(\mathrm{d}t-\alpha \sin^2 \theta \mathrm{d}\phi\right)^2 \frac{\Delta}{\rho^2}-\left( (r^2+\alpha^2)\mathrm{d}\phi -\alpha \mathrm{d}t\right)^2 \frac{\sin^2 \theta}{\rho^2}$$

The parameters $\alpha$ and $\rho$ depend on the angular momentum, and $\Delta$ in fact does depend on the charge of the black hole. Evidently, the curvature forms are also dependent on these.


Why isn't spacetime curved due to other forces or aspects of bodies?

Spacetime is affected by the presence of other fields. For example, electric and magnetic fields are described by the Maxwell Lagrangian,

$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

where $F=\mathrm{d}A$ is the field-strength, a closed $2$-form. The field theory has a non-vanishing stress-energy tensor (derived by applying Noether's theorem to spacetime translations) which sits on the right-hand side of the field equations, and will induce curvature. TheAnother example: the Kaluza-Klein metric in Kaluza-Klein theory is given by,

$$\mathrm{d}s^2 = g_{\mu\nu}\mathrm{d}x^\mu \mathrm{d}x^\nu -e^{2\sigma(x)}\left[ \mathrm{d}\psi + A_\mu \mathrm{d}x^\mu\right]^2$$ Hence, in this $5D$ model spacetime is influenced by a scalar field $\sigma(x)$, and a four-potential $A_\mu$.


For completeness, the action which the Kaluza-Klein metric gives rise to is,

$$S=-\frac{1}{16\pi G}\int \mathrm{d}^4 x \, \sqrt{g_4} \, \mathrm{d}\psi \,e^{\sigma} \left[ R^{(4)} + \frac{1}{4}e^{2\sigma}F_{\mu\nu}F^{\mu\nu} -2e^{-\sigma} \square e^{\sigma}\right]$$

which reduces to Einstein-Maxwell theory if $\psi \sim \psi + L$, for some period $L$ and the dilaton $\sigma=\mathrm{const}$.

Gravitation couples to anything within the stress-energy tensor, as dictated by the field equations,

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi G T_{\mu\nu}$$

Charge and angular momentum both affect the curvature of spacetime, as they affect the metric. For example, consider a spinning charged black hole, desribed by the Kerr-Newman metric,

$$\mathrm{d}s^2 = -\left( {\mathrm{d}r^2 \over \Delta + \mathrm{d}\theta^2} \right)\rho^2 + \left(\mathrm{d}t-\alpha \sin^2 \theta \mathrm{d}\phi\right)^2 \frac{\Delta}{\rho^2}-\left( (r^2+\alpha^2)\mathrm{d}\phi -\alpha \mathrm{d}t\right)^2 \frac{\sin^2 \theta}{\rho^2}$$

The parameters $\alpha$ and $\rho$ depend on the angular momentum, and $\Delta$ in fact does depend on the charge of the black hole. Evidently, the curvature forms are also dependent on these.


Why isn't spacetime curved due to other forces or aspects of bodies?

Spacetime is affected by the presence of other fields. For example, electric and magnetic fields are described by the Maxwell Lagrangian,

$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

where $F=\mathrm{d}A$ is the field-strength, a closed $2$-form. The field theory has a non-vanishing stress-energy tensor (derived by applying Noether's theorem to spacetime translations) which sits on the right-hand side of the field equations, and will induce curvature. The Kaluza-Klein metric in Kaluza-Klein theory is given by,

$$\mathrm{d}s^2 = g_{\mu\nu}\mathrm{d}x^\mu \mathrm{d}x^\nu -e^{2\sigma(x)}\left[ \mathrm{d}\psi + A_\mu \mathrm{d}x^\mu\right]^2$$ Hence, in this $5D$ model spacetime is influenced by a scalar field $\sigma(x)$, and a four-potential $A_\mu$.


For completeness, the action which the Kaluza-Klein metric gives rise to is,

$$S=-\frac{1}{16\pi G}\int \mathrm{d}^4 x \, \sqrt{g_4} \, \mathrm{d}\psi \,e^{\sigma} \left[ R^{(4)} + \frac{1}{4}e^{2\sigma}F_{\mu\nu}F^{\mu\nu} -2e^{-\sigma} \square e^{\sigma}\right]$$

which reduces to Einstein-Maxwell theory if $\psi \sim \psi + L$, for some period $L$ and the dilaton $\sigma=\mathrm{const}$.

Gravitation couples to anything within the stress-energy tensor, as dictated by the field equations,

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi G T_{\mu\nu}$$

Charge and angular momentum both affect the curvature of spacetime, as they affect the metric. For example, consider a spinning charged black hole, desribed by the Kerr-Newman metric,

$$\mathrm{d}s^2 = -\left( {\mathrm{d}r^2 \over \Delta + \mathrm{d}\theta^2} \right)\rho^2 + \left(\mathrm{d}t-\alpha \sin^2 \theta \mathrm{d}\phi\right)^2 \frac{\Delta}{\rho^2}-\left( (r^2+\alpha^2)\mathrm{d}\phi -\alpha \mathrm{d}t\right)^2 \frac{\sin^2 \theta}{\rho^2}$$

The parameters $\alpha$ and $\rho$ depend on the angular momentum, and $\Delta$ in fact does depend on the charge of the black hole. Evidently, the curvature forms are also dependent on these.


Why isn't spacetime curved due to other forces or aspects of bodies?

Spacetime is affected by the presence of other fields. For example, electric and magnetic fields are described by the Maxwell Lagrangian,

$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

where $F=\mathrm{d}A$ is the field-strength, a closed $2$-form. The field theory has a non-vanishing stress-energy tensor (derived by applying Noether's theorem to spacetime translations) which sits on the right-hand side of the field equations, and will induce curvature. Another example: the Kaluza-Klein metric in Kaluza-Klein theory is given by,

$$\mathrm{d}s^2 = g_{\mu\nu}\mathrm{d}x^\mu \mathrm{d}x^\nu -e^{2\sigma(x)}\left[ \mathrm{d}\psi + A_\mu \mathrm{d}x^\mu\right]^2$$ Hence, in this $5D$ model spacetime is influenced by a scalar field $\sigma(x)$, and a four-potential $A_\mu$.


For completeness, the action which the Kaluza-Klein metric gives rise to is,

$$S=-\frac{1}{16\pi G}\int \mathrm{d}^4 x \, \sqrt{g_4} \, \mathrm{d}\psi \,e^{\sigma} \left[ R^{(4)} + \frac{1}{4}e^{2\sigma}F_{\mu\nu}F^{\mu\nu} -2e^{-\sigma} \square e^{\sigma}\right]$$

which reduces to Einstein-Maxwell theory if $\psi \sim \psi + L$, for some period $L$ and the dilaton $\sigma=\mathrm{const}$.

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JamalS
  • 19.5k
  • 6
  • 59
  • 107

Gravitation couples to anything within the stress-energy tensor, as dictated by the field equations,

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi G T_{\mu\nu}$$

Charge and angular momentum both affect the curvature of spacetime, as they affect the metric. For example, consider a spinning charged black hole, desribed by the Kerr-Newman metric,

$$\mathrm{d}s^2 = -\left( {\mathrm{d}r^2 \over \Delta + \mathrm{d}\theta^2} \right)\rho^2 + \left(\mathrm{d}t-\alpha \sin^2 \theta \mathrm{d}\phi\right)^2 \frac{\Delta}{\rho^2}-\left( (r^2+\alpha^2)\mathrm{d}\phi -\alpha \mathrm{d}t\right)^2 \frac{\sin^2 \theta}{\rho^2}$$

The parameters $\alpha$ and $\rho$ depend on the angular momentum, and $\Delta$ in fact does depend on the charge of the black hole. Evidently, the curvature forms are also dependent on these.


Why isn't spacetime curved due to other forces or aspects of bodies?

Spacetime is affected by the presence of other fields. For example, electric and magnetic fields are described by the Maxwell Lagrangian,

$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

where $F=\mathrm{d}A$ is the field-strength, a closed $2$-form. The field theory has a non-vanishing stress-energy tensor (derived by applying Noether's theorem to spacetime translations) which sits on the right-hand side of the field equations, and will induce curvature. The Kaluza-Klein metric in Kaluza-Klein theory is given by,

$$\mathrm{d}s^2 = g_{\mu\nu}\mathrm{d}x^\mu \mathrm{d}x^\nu -e^{-2\sigma(x)}\left[ \mathrm{d}\psi + A_\mu \mathrm{d}x^\mu\right]^2$$$$\mathrm{d}s^2 = g_{\mu\nu}\mathrm{d}x^\mu \mathrm{d}x^\nu -e^{2\sigma(x)}\left[ \mathrm{d}\psi + A_\mu \mathrm{d}x^\mu\right]^2$$ Hence, in this $5D$ model spacetime is influenced by a scalar field $\sigma(x)$, and a four-potential $A_\mu$.


For completeness, the action which the Kaluza-Klein metric gives rise to is,

$$S=-\frac{1}{16\pi G}\int \mathrm{d}^4 x \, \sqrt{g_4} \, \mathrm{d}\psi \,e^{\sigma} \left[ R^{(4)} + \frac{1}{4}e^{2\sigma}F_{\mu\nu}F^{\mu\nu} -2e^{-\sigma} \square e^{\sigma}\right]$$

which reduces to Einstein-Maxwell theory if $\psi \sim \psi + L$, for some period $L$ and the dilaton $\sigma=\mathrm{const}$.

Gravitation couples to anything within the stress-energy tensor, as dictated by the field equations,

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi G T_{\mu\nu}$$

Charge and angular momentum both affect the curvature of spacetime, as they affect the metric. For example, consider a spinning charged black hole, desribed by the Kerr-Newman metric,

$$\mathrm{d}s^2 = -\left( {\mathrm{d}r^2 \over \Delta + \mathrm{d}\theta^2} \right)\rho^2 + \left(\mathrm{d}t-\alpha \sin^2 \theta \mathrm{d}\phi\right)^2 \frac{\Delta}{\rho^2}-\left( (r^2+\alpha^2)\mathrm{d}\phi -\alpha \mathrm{d}t\right)^2 \frac{\sin^2 \theta}{\rho^2}$$

The parameters $\alpha$ and $\rho$ depend on the angular momentum, and $\Delta$ in fact does depend on the charge of the black hole. Evidently, the curvature forms are also dependent on these.


Why isn't spacetime curved due to other forces or aspects of bodies?

Spacetime is affected by the presence of other fields. For example, electric and magnetic fields are described by the Maxwell Lagrangian,

$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

where $F=\mathrm{d}A$ is the field-strength, a closed $2$-form. The field theory has a non-vanishing stress-energy tensor (derived by applying Noether's theorem to spacetime translations) which sits on the right-hand side of the field equations, and will induce curvature. The Kaluza-Klein metric in Kaluza-Klein theory is given by,

$$\mathrm{d}s^2 = g_{\mu\nu}\mathrm{d}x^\mu \mathrm{d}x^\nu -e^{-2\sigma(x)}\left[ \mathrm{d}\psi + A_\mu \mathrm{d}x^\mu\right]^2$$ Hence, in this $5D$ model spacetime is influenced by a scalar field $\sigma(x)$, and a four-potential $A_\mu$.

Gravitation couples to anything within the stress-energy tensor, as dictated by the field equations,

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi G T_{\mu\nu}$$

Charge and angular momentum both affect the curvature of spacetime, as they affect the metric. For example, consider a spinning charged black hole, desribed by the Kerr-Newman metric,

$$\mathrm{d}s^2 = -\left( {\mathrm{d}r^2 \over \Delta + \mathrm{d}\theta^2} \right)\rho^2 + \left(\mathrm{d}t-\alpha \sin^2 \theta \mathrm{d}\phi\right)^2 \frac{\Delta}{\rho^2}-\left( (r^2+\alpha^2)\mathrm{d}\phi -\alpha \mathrm{d}t\right)^2 \frac{\sin^2 \theta}{\rho^2}$$

The parameters $\alpha$ and $\rho$ depend on the angular momentum, and $\Delta$ in fact does depend on the charge of the black hole. Evidently, the curvature forms are also dependent on these.


Why isn't spacetime curved due to other forces or aspects of bodies?

Spacetime is affected by the presence of other fields. For example, electric and magnetic fields are described by the Maxwell Lagrangian,

$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

where $F=\mathrm{d}A$ is the field-strength, a closed $2$-form. The field theory has a non-vanishing stress-energy tensor (derived by applying Noether's theorem to spacetime translations) which sits on the right-hand side of the field equations, and will induce curvature. The Kaluza-Klein metric in Kaluza-Klein theory is given by,

$$\mathrm{d}s^2 = g_{\mu\nu}\mathrm{d}x^\mu \mathrm{d}x^\nu -e^{2\sigma(x)}\left[ \mathrm{d}\psi + A_\mu \mathrm{d}x^\mu\right]^2$$ Hence, in this $5D$ model spacetime is influenced by a scalar field $\sigma(x)$, and a four-potential $A_\mu$.


For completeness, the action which the Kaluza-Klein metric gives rise to is,

$$S=-\frac{1}{16\pi G}\int \mathrm{d}^4 x \, \sqrt{g_4} \, \mathrm{d}\psi \,e^{\sigma} \left[ R^{(4)} + \frac{1}{4}e^{2\sigma}F_{\mu\nu}F^{\mu\nu} -2e^{-\sigma} \square e^{\sigma}\right]$$

which reduces to Einstein-Maxwell theory if $\psi \sim \psi + L$, for some period $L$ and the dilaton $\sigma=\mathrm{const}$.

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JamalS
  • 19.5k
  • 6
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  • 107

Gravitation couples to anything within the stress-energy tensor, as dictated by the field equations,

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi G T_{\mu\nu}$$

Charge and angular momentum both affect the curvature of spacetime, as they affect the metric. For example, consider a spinning charged black hole, desribed by the Kerr-Newman metric,

$$\mathrm{d}s^2 = -\left( {\mathrm{d}r^2 \over \Delta + \mathrm{d}\theta^2} \right)\rho^2 + \left(\mathrm{d}t-\alpha \sin^2 \theta \mathrm{d}\phi\right)^2 \frac{\Delta}{\rho^2}-\left( (r^2+\alpha^2)\mathrm{d}\phi -\alpha \mathrm{d}t\right)^2 \frac{\sin^2 \theta}{\rho^2}$$

The parameters $\alpha$ and $\rho$ depend on the angular momentum, and $\Delta$ in fact does depend on the charge of the black hole. Evidently, the curvature forms are also dependent on these.


Why isn't spacetime curved due to other forces or aspects of bodies?

Spacetime is affected by the presence of other fields. For example, electric and magnetic fields are described by the Maxwell Lagrangian,

$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

where $F=\mathrm{d}A$ is the field-strength, a closed $2$-form. The field theory has a non-vanishing stress-energy tensor (derived by applying Noether's theorem to spacetime translations) which sits on the right-hand side of the field equations, and will induce curvature. The Kaluza-Klein metric in Kaluza-Klein theory is given by,

$$\mathrm{d}s^2 = g_{\mu\nu}\mathrm{d}x^\mu \mathrm{d}x^\nu -e^{-2\sigma(x)}\left[ \mathrm{d}\psi + A_\mu \mathrm{d}x^\mu\right]^2$$ Hence, in this $5D$ model spacetime is influenced by a scalar field $\sigma(x)$, and a four-potential $A_\mu$.

Gravitation couples to anything within the stress-energy tensor, as dictated by the field equations,

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi G T_{\mu\nu}$$

Charge and angular momentum both affect the curvature of spacetime, as they affect the metric. For example, consider a spinning charged black hole, desribed by the Kerr-Newman metric,

$$\mathrm{d}s^2 = -\left( {\mathrm{d}r^2 \over \Delta + \mathrm{d}\theta^2} \right)\rho^2 + \left(\mathrm{d}t-\alpha \sin^2 \theta \mathrm{d}\phi\right)^2 \frac{\Delta}{\rho^2}-\left( (r^2+\alpha^2)\mathrm{d}\phi -\alpha \mathrm{d}t\right)^2 \frac{\sin^2 \theta}{\rho^2}$$

The parameters $\alpha$ and $\rho$ depend on the angular momentum, and $\Delta$ in fact does depend on the charge of the black hole. Evidently, the curvature forms are also dependent on these.

Gravitation couples to anything within the stress-energy tensor, as dictated by the field equations,

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi G T_{\mu\nu}$$

Charge and angular momentum both affect the curvature of spacetime, as they affect the metric. For example, consider a spinning charged black hole, desribed by the Kerr-Newman metric,

$$\mathrm{d}s^2 = -\left( {\mathrm{d}r^2 \over \Delta + \mathrm{d}\theta^2} \right)\rho^2 + \left(\mathrm{d}t-\alpha \sin^2 \theta \mathrm{d}\phi\right)^2 \frac{\Delta}{\rho^2}-\left( (r^2+\alpha^2)\mathrm{d}\phi -\alpha \mathrm{d}t\right)^2 \frac{\sin^2 \theta}{\rho^2}$$

The parameters $\alpha$ and $\rho$ depend on the angular momentum, and $\Delta$ in fact does depend on the charge of the black hole. Evidently, the curvature forms are also dependent on these.


Why isn't spacetime curved due to other forces or aspects of bodies?

Spacetime is affected by the presence of other fields. For example, electric and magnetic fields are described by the Maxwell Lagrangian,

$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

where $F=\mathrm{d}A$ is the field-strength, a closed $2$-form. The field theory has a non-vanishing stress-energy tensor (derived by applying Noether's theorem to spacetime translations) which sits on the right-hand side of the field equations, and will induce curvature. The Kaluza-Klein metric in Kaluza-Klein theory is given by,

$$\mathrm{d}s^2 = g_{\mu\nu}\mathrm{d}x^\mu \mathrm{d}x^\nu -e^{-2\sigma(x)}\left[ \mathrm{d}\psi + A_\mu \mathrm{d}x^\mu\right]^2$$ Hence, in this $5D$ model spacetime is influenced by a scalar field $\sigma(x)$, and a four-potential $A_\mu$.

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JamalS
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